Chapter 4 Resource Masters - KTL MATH CLASSES©Glencoe/McGraw-Hill iv Glencoe Algebra 2 Teacher’s Guide to Using the Chapter 4 Resource Masters The Fast FileChapter Resource system

Chapter 4Resource Masters

Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.

Study Guide and Intervention Workbook 0-07-828029-XSkills Practice Workbook 0-07-828023-0Practice Workbook 0-07-828024-9

ANSWERS FOR WORKBOOKS The answers for Chapter 4 of these workbookscan be found in the back of this Chapter Resource Masters booklet.

Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teacher, and families without charge; and be used solely in conjunction with Glencoe’s Algebra 2. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.

Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 0-07-828007-9 Algebra 2Chapter 4 Resource Masters

1 2 3 4 5 6 7 8 9 10 066 11 10 09 08 07 06 05 04 03 02

Glenc oe/ M cGr aw-H ill

© Glencoe/McGraw-Hill iii Glencoe Algebra 2

Contents

Vocabulary Builder . . . . . . . . . . . . . . . . vii

Lesson 4-1Study Guide and Intervention . . . . . . . . 169–170Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 171Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 172Reading to Learn Mathematics . . . . . . . . . . 173Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 174








Chapter 4 AssessmentChapter 4 Test, Form 1 . . . . . . . . . . . . 217–218Chapter 4 Test, Form 2A . . . . . . . . . . . 219–220Chapter 4 Test, Form 2B . . . . . . . . . . . 221–222Chapter 4 Test, Form 2C . . . . . . . . . . . 223–224Chapter 4 Test, Form 2D . . . . . . . . . . . 225–226Chapter 4 Test, Form 3 . . . . . . . . . . . . 227–228Chapter 4 Open-Ended Assessment . . . . . . 229Chapter 4 Vocabulary Test/Review . . . . . . . 230Chapter 4 Quizzes 1 & 2 . . . . . . . . . . . . . . . 231Chapter 4 Quizzes 3 & 4 . . . . . . . . . . . . . . . 232Chapter 4 Mid-Chapter Test . . . . . . . . . . . . 233Chapter 4 Cumulative Review . . . . . . . . . . . 234Chapter 4 Standardized Test Practice . . 235–236Unit 1 Test/Review (Ch. 1–4) . . . . . . . . 237–238

Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A36

© Glencoe/McGraw-Hill iv Glencoe Algebra 2

Teacher’s Guide to Using theChapter 4 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 4 Resource Masters includes the core materials neededfor Chapter 4. These materials include worksheets, extensions, and assessment options.The answers for these pages appear at the back of this booklet.

All of the materials found in this booklet are included for viewing and printing in theAlgebra 2 TeacherWorks CD-ROM.

Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.

WHEN TO USE Give these pages tostudents before beginning Lesson 4-1.Encourage them to add these pages to theirAlgebra 2 Study Notebook. Remind them to add definitions and examples as theycomplete each lesson.

Study Guide and InterventionEach lesson in Algebra 2 addresses twoobjectives. There is one Study Guide andIntervention master for each objective.

WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.

Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.

WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.

Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.

WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.

Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.

WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.

Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.

WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.

© Glencoe/McGraw-Hill v Glencoe Algebra 2

Assessment OptionsThe assessment masters in the Chapter 4Resource Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.

Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions

and is intended for use with basic levelstudents.

• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.

• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.

• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.

All of the above tests include a free-response Bonus question.

• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.

• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.

Intermediate Assessment• Four free-response quizzes are included

to offer assessment at appropriateintervals in the chapter.

• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.

Continuing Assessment• The Cumulative Review provides

students an opportunity to reinforce andretain skills as they proceed throughtheir study of Algebra 2. It can also beused as a test. This master includes free-response questions.

• The Standardized Test Practice offerscontinuing review of algebra concepts invarious formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and quantitative-comparison questions. Bubble-in andgrid-in answer sections are provided onthe master.

Answers• Page A1 is an answer sheet for the

Standardized Test Practice questionsthat appear in the Student Edition onpages 216–217. This improves students’familiarity with the answer formats theymay encounter in test taking.

• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.

• Full-size answer keys are provided forthe assessment masters in this booklet.

Reading to Learn MathematicsVocabulary Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

44

Voca

bula

ry B

uild

erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 4.As you study the chapter, complete each term’s definition or description.Remember to add the page number where you found the term. Add these pages toyour Algebra Study Notebook to review vocabulary at the end of the chapter.

Vocabulary Term Found on Page Definition/Description/Example

Cramer’s Rule

KRAY·muhrs

determinant

dilation

dy·LAY·shuhn

element

expansion by minors

identity matrix

image

inverse

isometry

eye·SAH·muh·tree

matrix

MAY·trihks

(continued on the next page)

© Glencoe/McGraw-Hill vii Glencoe Algebra 2

© Glencoe/McGraw-Hill viii Glencoe Algebra 2

Vocabulary Term Found on Page Definition/Description/Example

matrix equation

preimage

reflection

rotation

scalar multiplication

SKAY·luhr

square matrix

transformation

translation

vertex matrix

zero matrix

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

44

Study Guide and InterventionIntroduction to Matrices

NAME ______________________________________________ DATE ____________ PERIOD _____

4-14-1

© Glencoe/McGraw-Hill 169 Glencoe Algebra 2

Less

on

4-1

Organize Data

Matrixa rectangular array of variables or constants in horizontal rows and vertical columns, usually enclosed in brackets.

A matrix can be described by its dimensions. A matrix with m rows and n columns is an m � n matrix.

Owls’ eggs incubate for 30 days and their fledgling period is also 30 days. Swifts’ eggs incubate for 20 days and their fledgling period is 44 days.Pigeon eggs incubate for 15 days, and their fledgling period is 17 days. Eggs of theking penguin incubate for 53 days, and the fledgling time for a king penguin is360 days. Write a 2 � 4 matrix to organize this information. Source: The Cambridge Factfinder

What are the dimensions of matrix A if A � ?

Since matrix A has 2 rows and 4 columns, the dimensions of A are 2 � 4.

State the dimensions of each matrix.

1. 4 � 4 2. [16 12 0] 1 � 3 3. 5 � 2

4. A travel agent provides for potential travelers the normal high temperatures for themonths of January, April, July, and October for various cities. In Boston these figures are36°, 56°, 82°, and 63°. In Dallas they are 54°, 76°, 97°, and 79°. In Los Angeles they are68°, 72°, 84°, and 79°. In Seattle they are 46°, 58°, 74°, and 60°, and in St. Louis they are38°, 67°, 89°, and 69°. Organize this information in a 4 � 5 matrix. Source: The New York Times Almanac

Boston Dallas Los Angeles Seattle St. LouisJanuary 36 54 68 46 38

April 56 76 72 58 67 July 82 97 84 74 89

October 63 79 79 60 69

71 44 39 27 45 16 92 53 78 65

15 5 27 �4 23 6 0 5 14 70 24 �3 63 3 42 90

13 10 �3 45 2 8 15 80

Owl Swift Pigeon King PenguinIncubation 30 20 15 53

Fledgling 30 44 17 360

Example 1Example 1

Example 2Example 2

ExercisesExercises


Equations Involving Matrices

Equal MatricesTwo matrices are equal if they have the same dimensions and each element of one matrix isequal to the corresponding element of the other matrix.

You can use the definition of equal matrices to solve matrix equations.

Solve � for x and y.

Since the matrices are equal, the corresponding elements are equal. When you write thesentences to show the equality, two linear equations are formed.4x � �2y � 2y � x � 8

This system can be solved using substitution.4x � �2y � 2 First equation

4x � �2(x � 8) � 2 Substitute x � 8 for y.

4x � �2x � 16 � 2 Distributive Property

6x � 18 Add 2x to each side.

x � 3 Divide each side by 6.

To find the value of y, substitute 3 for x in either equation.y � x � 8 Second equation

y � 3 � 8 Substitute 3 for x.

y � �5 Subtract.

The solution is (3, �5).

Solve each equation.

1. [5x 4y] � [20 20] 2. � 3. �

(4, 5) � , �6� (�2, �7)

4. � 5. � 6. �

(4, 2.5) (6, �3) (�5, 8)

7. � 8. � 9. �

� , �8� �� , � � (18, 21)

10. � 11. � 12. �

(2, 5.2) (2.5, 4) (�12.5, �12.5)

0 �25

x � y x � y

17 �8

2x � 3y �4x � 0.5y

7.5 8.0

3x � 1.5 2y � 2.4

1�2

3�4

5�4

9 51

�3x

� � �7y

� �2

x� � 2y

�3 �11

8x � 6y 12x � 4y

18 20 12 �13

8x � y 16x 12 y � 4x

�1 �18

5x � 3y 2x � y

3 12

2x � 3y x � 2y

�1 22

x � 2y 3x � 4y

4�3

4 � 5x y � 5

�2y x

28 � 4y �3x � 2

3x y

�2y � 2 x � 8

4x y

Study Guide and Intervention (continued)

Introduction to Matrices

NAME ______________________________________________ DATE ____________ PERIOD _____

4-14-1

ExampleExample

ExercisesExercises

Skills PracticeIntroduction to Matrices

NAME ______________________________________________ DATE ____________ PERIOD _____

4-14-1


Less

on

4-1


1. 2 � 3 2. [0 15] 1 � 2

3. 2 � 2 4. 3 � 3

5. 2 � 4 6. 4 � 1


7. [5x 3y] � [15 12] (3, 4) 8. [3x � 2] � [7] 3

9. � (�2, 7) 10. [2x �8y z] � [10 16 �1] (5, �2, �1)

11. � (4, 5) 12. � (2, 4)

13. � (�4, 3) 14. � (0, �2)

15. � (1, �1) 16. � (2, 11, 7)

17. � (9, 4, 3) 18. � (1, 4, 0)

19. � (3, 1) 20. � (12, 3) 4y x � 3

x 3y

6y x

2x y � 2

4x � 1 13 4z

5x 4y � 3 8z

9 4y 9

x 16 3z

7 2y � 4 12 28

3x � 1 18 12 4z

3x y � 3

4x � 1 9y � 5

5x � 2 3y � 10

3x � 2 7y � 2

�20 8y

5x 24

10x 32

20 56 � 6y

4 2

8 � x2y � 8

�14 2y

7x 14

�1 �1 �1 �3

9 3 �3 �6 3 4 �4 5

6 1 2 �3 4 5 �2 7 9

3 2 1 8

3 2 4 �1 4 0



1. [�3 �3 7] 1 � 3 2. 2 � 3 3. 3 � 4


4. [4x 42] � [24 6y] (6, 7)

5. [�2x 22 �3z] � [6x �2y 45] (0, �11, �15)

6. � (�6, 7) 7. � (4, 1)

8. � (�3, 2) 9. � (�1, 1)

10. � (�1, �1, 1) 11. � (7, 13)

12. � (3, �2) 13. � (2, �4)

14. TICKET PRICES The table at the right gives ticket prices for a concert. Write a 2 � 3 matrixthat represents the cost of a ticket.

CONSTRUCTION For Exercises 15 and 16, use the following information.During each of the last three weeks, a road-building crew has used three truck-loads of gravel. The table at the right shows theamount of gravel in each load.

15. Write a matrix for the amount of gravel in each load.

or

16. What are the dimensions of the matrix? 3 � 3

40 40 3232 40 2424 32 24

40 32 2440 40 3232 24 24

Week 1 Week 2 Week 3

Load 1 40 tons Load 1 40 tons Load 1 32 tons



6 12 188 15 22

Child Student Adult

Cost Purchasedin Advance

$6 $12 $18

Cost Purchasedat the Door

$8 $15 $22

0 �2

2x � y 3x � 2y

�1 y

5x � 8y 3x � 11

y � 8 17

3x y � 4

�5 x � 1 3y 5z � 4

�5 3x � 1 2y � 1 3z � 2

�3x � 21 8y � 5

6x � 12 �3y � 6

�3x �2y � 16

�4x � 3 6y

20 2y � 3

7x � 8 8y � 3

�36 17

6x 2y � 3

�2 2 �2 3 5 16 0 0 4 7 �1 4

5 8 �1 �2 1 8

Practice (Average)

Introduction to Matrices

NAME ______________________________________________ DATE ____________ PERIOD _____

4-14-1

Reading to Learn MathematicsIntroduction to Matrices

NAME ______________________________________________ DATE ____________ PERIOD _____

4-14-1


Less

on

4-1

Pre-Activity How are matrices used to make decisions?

Read the introduction to Lesson 4-1 at the top of page 154 in your textbook.

What is the base price of a Mid-Size SUV?$27,975

Reading the Lesson

1. Give the dimensions of each matrix.

a. 2 � 3 b. [1 4 0 �8 2] 1 � 5

2. Identify each matrix with as many of the following descriptions that apply: row matrix,column matrix, square matrix, zero matrix.

a. [6 5 4 3] row matrix

b. column matrix; zero matrix

c. [0] row matrix; column matrix; square matrix; zero matrix

3. Write a system of equations that you could use to solve the following matrix equation forx, y, and z. (Do not actually solve the system.)

� 3x � �9, x � y � 5, y � z � 6

Helping You Remember

4. Some students have trouble remembering which number comes first in writing thedimensions of a matrix. Think of an easy way to remember this.

Sample answer: Read the matrix from top to bottom, then from left toright. Reading down gives the number of rows, which is written first inthe dimensions of the matrix. Reading across gives the number ofcolumns, which is written second.

�9 5 6

3x x � y y � z

0 0 0

3 2 5 �1 0 6


TessellationsA tessellation is an arangement of polygons covering a plane without any gaps or overlapping. One example of a tessellation is a honeycomb. Three congruent regular hexagons meet at each vertex, and there is no wasted space between cells. Thistessellation is called a regular tessellation since it is formed by congruent regular polygons.

A semi-regular tessellation is a tessellation formed by two or more regular polygons such that the number of sides of the polygons meeting at each vertex is the same.

For example, the tessellation at the left has two regulardodecagons and one equilateral triangle meeting at eachvertex. We can name this tessellation a 3-12-12 for thenumber of sides of each polygon that meet at one vertex.

Name each semi-regular tessellation shown according to the number of sides of the polygons that meet at each vertex.

1. 2.

3-3-3-3-6 4-6-12

An equilateral triangle, two squares, and a regular hexagon can be used to surround a point in two different orders. Continue each pattern to see which is a semi-regular tessellation.

3. 3-4-4-6 4. 3-4-6-4

not semi-regular semi-regular

On another sheet of paper, draw part of each design. Then determine if it is a semi-regular tessellation.

5. 3-3-4-12 6. 3-4-3-12 7. 4-8-8 8. 3-3-3-4-4not semi-regular not semi-regular semi-regular semi-regular

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

4-14-1

Study Guide and InterventionOperations with Matrices

NAME ______________________________________________ DATE ____________ PERIOD _____

4-24-2


Less

on

4-2

Add and Subtract Matrices

Addition of Matrices � �

Subtraction of Matrices � �

Find A � B if A � and B � .

A � B � �

�

�

Find A � B if A � and B � .

A � B � �

� �

Perform the indicated operations. If the matrix does not exist, write impossible.

1. � 2. �

3. � [�6 3 �2] impossible 4. �

5. � 6. �

�14

� ��145�

��

76

� �161�

10 �1 �13 1 9 �14 1 �2 �2

�12� �

23�

�23� ��

12�

�34� �

25�

��12� �

43�

�2 1 7 3 �4 3 �8 5 6

8 0 �6 4 5 �11 �7 3 4

�6 4�2 1 11 2

�11 6 2 �5 4 �7

5 �2 �4 6 7 9

6 �3 2

2 �2 11 3 13 1

�4 3 2 6 9 �4

6 �5 9 �3 4 5

12 4�12 6

�4 3 2 �12

8 7 �10 �6

�6 11 5 �5 16 �1

�2 � 4 8 � (�3) 3 � (�2) �4 � 1 10 � (�6) 7 � 8

4 �3 �2 1 �6 8

�2 8 3 �4 10 7

4 �3 �2 1 �6 8

�2 8 3 �4 10 7

10 �5 �3 �18

6 � 4 �7 � 2 2 � (�5) �12 � (�6)

4 2 �5 �6

6 �7 2 �12

4 2�5 �6

6 �72 �12

a � j b � k c � l d � m e � n f � o g � p h � q i � r

j k l m n o p q r

a b c d e f g h i

a � j b � k c � l d � m e � n f � o g � p h � q i � r

j k l m n o p q r

a b c d e f g h i

Example 1Example 1

Example 2Example 2

ExercisesExercises

Scalar Multiplication You can multiply an m � n matrix by a scalar k.

Scalar Multiplication k �

If A � and B � , find 3B � 2A.

3B � 2A � 3 � 2 Substitution

� � Multiply.

� � Simplify.

� Subtract.

� Simplify.

Perform the indicated matrix operations. If the matrix does not exist, writeimpossible.

1. 6 2. � 3. 0.2

4. 3 � 2 5. �2 � 4

6. 2 � 5 7. 4 � 2

8. 8 � 3 9. � � � 3 �1 ��

32

� �74

�

3 �5 1 7

9 1 �7 0

1�4

28 8 18 1�7 20

4 0 �2 3 3 �4

2 1 3 �1 �2 4

�4 �14 28�16 26 6

4 3 �4 2 �5 �1

1 �2 5 �3 4 1

22 �1510 31

2 1 4 3

6 �10 �5 8

�14 2 8 6

�2 0 2 5

3 �1 0 7

�10 11 12 �1

�1 2 �3 5

�4 5 2 3

5 �2 �9 1 11 �612 7 �19

�2 �5 �3�17 11 �8 6 �1 �15

12 �30 18 0 42 �6�24 36 54

25 �10 �45 5 55 �30 60 35 �95

6 15 9 51 �33 24 �18 3 45

1�3

2 �5 3 0 7 �1 �4 6 9

�11 15 33 18

�3 � 8 15 � 0 21 � (�12) 24 � 6

8 0 �12 6

�3 15 21 24

2(4) 2(0) 2(�6) 2(3)

3(�1) 3(5) 3(7) 3(8)

4 0 �6 3

�1 5 7 8

�1 5 7 8

4 0�6 3

ka kb kc kd ke kf

a b c d e f



Operations with Matrices

NAME ______________________________________________ DATE ____________ PERIOD _____

4-24-2

ExampleExample

ExercisesExercises

Skills PracticeOperations with Matrices

NAME ______________________________________________ DATE ____________ PERIOD _____

4-24-2


Less

on

4-2


1. [5 �4] � [4 5] [9 1] 2. �

3. [3 1 6] � impossible 4. �

5. 3[9 4 �3] [27 12 �9] 6. [6 �3] � 4[4 7] [�10 �31]

7. �2 � 8. 3 � 4

9. 5 � 2 10. 3 � 2

Use A � , B � , and C � to find the following.

11. A � B 12. B � C

13. B � A 14. A � B � C

15. 3B 16. �5C

17. A � 4C 18. 2B � 3A 13 1014 5

15 �14�8 �1

15 �20�15 �5

6 63 �6

2 88 2

�1 0�3 �5

5 �2�2 �3

5 45 1

�3 4 3 1

2 21 �2

3 24 3

7 5 �1�24 9 21

1 �1 5 6 6 �3

3 1 3 �4 7 5

�8 40 44 1�3 5

6 5 �3 �2 1 0

�4 6 10 1 �1 1

16 �8�49

2 2 10

8 0 �3

5 �9�9 �17

1 1 1 1

�2 5 5 9

14 8 4 5 14 �2

9 9 2 4 6 4

5 �1 2 1 8 �6

4 �1 2

8 10�7 �3

0 �7 6 2

8 3 �1 �1



1. � 2. �

3. �3 � 4 4. 7 � 2

5. �2 � 4 � 6. �

Use A � , B � , and C � to find the following.

7. A � B 8. A � C

9. �3B 10. 4B � A

11. �2B � 3C 12. A � 0.5C

ECONOMICS For Exercises 13 and 14, use the table that shows loans by an economicdevelopment board to women and men starting new businesses.

13. Write two matrices that represent the number of newbusinesses and loan amounts,one for women and one for men.

,

14. Find the sum of the numbers of new businesses and loan amounts for both men and women over the three-year period expressed as a matrix.

15. PET NUTRITION Use the table that gives nutritional information for two types of dog food. Find thedifference in the percent of protein, fat, and fiberbetween Mix B and Mix A expressed as a matrix.[2 �4 3]

% Protein % Fat % Fiber

Mix A 22 12 5

Mix B 24 8 8

63 1,431,00073 1,574,00063 1,339,000

36 864,00032 672,00028 562,000

27 567,00041 902,00035 777,000

Women Men

BusinessesLoan

BusinessesLoan

Amount ($) Amount ($)

1999 27 $567,000 36 $864,000

2000 41 $902,000 32 $672,000

2001 35 $777,000 28 $562,000

9 �5 3�6 4 12

�26 16 �28 16 12 �42

�12 17 20 7 �6 �38

6 �12 �15�3 0 27

�6 7 �6 3 10 �18

6 �5 �5�4 6 11

10 �8 6�6 �4 20

�2 4 5 1 0 �9

4 �1 0�3 6 2

24 324 3

27 �9 54 �18

2�3

8 12 �16 20

3�4

�12 �2

10 18

0 5

1 2

�1 4 �3 7 2 �6

2 �1 8 4 7 9

�9 64�135 81

�3 16 �21 12

�1 0 17 �11

71�116 42

�67 45 �24

4 �71 18

�4 8 10 �4 6 8

�6 9 7 �11 �8 17

2 �1 3 7 14 �9

Practice (Average)

Operations with Matrices

NAME ______________________________________________ DATE ____________ PERIOD _____

4-24-2

16 �15 6214 45 75

Reading to Learn MathematicsOperations with Matrices

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Pre-Activity How can matrices be used to calculate daily dietary needs?


• Write a sum that represents the total number of Calories in the patient’sdiet for Day 2. (Do not actually calculate the sum.) 482 � 622 � 987

• Write the sum that represents the total fat content in the patient’s dietfor Day 3. (Do not actually calculate the sum.) 11 � 12 � 38

Reading the Lesson

1. For each pair of matrices, give the dimensions of the indicated sum, difference, or scalarproduct. If the indicated sum, difference, or scalar product does not exist, write impossible.

A � B �

C � D �

A � D: C � D: 5B:

�4C: 2D � 3A:

2. Suppose that M, N, and P are nonzero 2 � 4 matrices and k is a negative real number.Indicate whether each of the following statements is true or false.

a. M � (N � P) � M � (P � N) true b. M � N � N � M false

c. M � (N � P) � (M � N) � P false d. k(M � N) � kM � kN true


3. The mathematical term scalar may be unfamiliar, but its meaning is related to the wordscale as used in a scale of miles on a map. How can this usage of the word scale help youremember the meaning of scalar?

Sample answer: A scale of miles tells you how to multiply the distancesyou measure on a map by a certain number to get the actual distancebetween two locations. This multiplier is often called a scale factor. Ascalar represents the same idea: It is a real number by which a matrixcan be multiplied to change all the elements of the matrix by a uniformscale factor.

2 � 33 � 2

2 � 2impossible2 � 3

�3 6 0 �8 4 0

5 10 �3 6 4 12

�4 0 0 �5

3 5 6 �2 8 1


Sundaram’s SieveThe properties and patterns of prime numbers have fascinated many mathematicians.In 1934, a young East Indian student named Sundaram constructed the following matrix.

4 7 10 13 16 19 22 25 . . .7 12 17 22 27 32 37 42 . . .

10 17 24 31 38 45 52 59 . . .13 22 31 40 49 58 67 76 . . .16 27 38 49 60 71 82 93 . . .. . . . . . . . . . .

A surprising property of this matrix is that it can be used to determine whether or not some numbers are prime.

Complete these problems to discover this property.

1. The first row and the first column are created by using an arithmetic pattern. What is the common difference used in the pattern?

2. Find the next four numbers in the first row.

3. What are the common differences used to create the patterns in rows 2, 3,4, and 5?

4. Write the next two rows of the matrix. Include eight numbers in each row.

5. Choose any five numbers from the matrix. For each number n, that you chose from the matrix, find 2n � 1.

6. Write the factorization of each value of 2n � 1 that you found in problem 5.

7. Use your results from problems 5 and 6 to complete this statement: If n

occurs in the matrix, then 2n � 1 (is/is not) a prime number.

8. Choose any five numbers that are not in the matrix. Find 2n � 1 for each of these numbers. Show that each result is a prime number.

9. Complete this statement: If n does not occur in the matrix, then 2n � 1 is

.a prime number

Enrichment

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Study Guide and InterventionMultiplying Matrices

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Multiply Matrices You can multiply two matrices if and only if the number of columnsin the first matrix is equal to the number of rows in the second matrix.

Multiplication of Matrices � �

Find AB if A � and B � .

AB � � Substitution

� Multiply columns by rows.

� Simplify.

Find each product, if possible.

1. � 2. � 3. �

4. � 5. � 6. �

7. � [0 4 �3] 8. � 9. �

not possible10 �1618 �6 5 9

11 �2113 �15

2 �2 3 1 �2 4

2 0 �3 1 4 �2 �1 3 1

1 �3 �2 0

7 �2 5 �4

6 10 �4 3 �2 7

24 �1514 �17

�1 4 8 4�3 �9

�15 1 7 26 �2 �12

4 �1 �2 5

5 �2 2 �3

1 2 2 1

3 �2 0 4 �5 1

4 0 �2 �3 1 1

�3 1 5 �2

7 �714 14

�3 �2 2 34

12 3�6 9

3 �1 2 4

3 �1 2 4

3 2 �1 4

�1 0 3 7

3 0 0 3

4 1 �2 3

�23 17 12 �10 �2 19

�4(5) � 3(�1) �4(�2) � 3(3) 2(5) � (�2)(�1) 2(�2) � (�2)(3) 1(5) � 7(�1) 1(�2) � 7(3)

5 �2 �1 3

�4 3 2 �2 1 7

5 �2�1 3

�4 3 2 �2 1 7

a1x1 � b1x2 a1y1 � b1y2 a2x1 � b2x2 a2y1 � b2y2

x1 y1 x2 y2

a1 b1 a2 b2

ExampleExample

ExercisesExercises


Multiplicative Properties The Commutative Property of Multiplication does not holdfor matrices.

Properties of Matrix MultiplicationFor any matrices A, B, and C for which the matrix product isdefined, and any scalar c, the following properties are true.

Associative Property of Matrix Multiplication (AB)C � A(BC)

Associative Property of Scalar Multiplication c(AB) � (cA)B � A(cB)

Left Distributive Property C(A � B) � CA � CB

Right Distributive Property (A � B)C � AC � BC

Use A � , B � , and C � to find each product.

a. (A � B)C

(A � B)C � � � � �

� �

�

�

b. AC � BC

AC � BC � � � �

� �

� � �

Note that although the results in the example illustrate the Right Distributive Property,they do not prove it.

Use A � , B � , C � , and scalar c � �4 to determine whether

each of the following equations is true for the given matrices.

1. c(AB) � (cA)B yes 2. AB � BA no

3. BC � CB no 4. (AB)C � A(BC) yes

5. C(A � B) � AC � BC no 6. c(A � B) � cA � cB yes

��12� �2

1 �3

6 42 1

3 25 �2

�12 �21 �5 �20

2 �4 �13 �19

�14 �17 8 �1

2(1) � 0(6) 2(�2) � 0(3) 5(1) � (�3)(6) 5(�2) � (�3)(3)

4(1) � (�3)(6) 4(�2) � (�3)(3) 2(1) � 1(6) 2(�2) � 1(3)

1 �2 6 3

2 0 5 �3

1 �2 6 3

4 �3 2 1

�12 �21 �5 �20

6(1) � (�3)(6) 6(�2) � (�3)(3) 7(1) � (�2)(6) 7(�2) � (�2)(3)

1 �2 6 3

6 �3 7 �2

1 �2 6 3

2 0 5 �3

4 �3 2 1

1 �26 3

2 05 �3

4 �32 1


Multiplying Matrices

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ExampleExample

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Determine whether each matrix product is defined. If so, state the dimensions ofthe product.

1. A2 � 5 � B5 � 1 2 � 1 2. M1 � 3 � N3 � 2 1 � 2

3. B3 � 2 � A3 � 2 undefined 4. R4 � 4 � S4 � 1 4 � 1

5. X3 � 3 � Y3 � 4 3 � 4 6. A6 � 4 � B4 � 5 6 � 5


7. [3 2] � [8] 8. �

9. � 10. � not possible

11. [�3 4] � [8 11] 12. � [2 �3 �2]

13. � not possible 14. �

15. � 16. �

Use A � , B � , C � , and scalar c � 2 to determine whether the following equations are true for the given matrices.

17. (AC)c � A(Cc) yes 18. AB � BA no

19. B(A � C) � AB � BC no 20. (A � B)c � Ac � Bc yes

3 �11 0

�3 2 5 1

2 12 1

44

2 2 2

0 1 1 1 1 0

�12 20 �6 8 6 0

3 �3 0 2

�4 4 �2 1 2 3

�6 6 15 12 3 �9

0 3 3 0

2 �2 4 5 �3 1

48

5 6 �3

�2 3 2 6 �9 �6

�1 3

0 �1 2 2

1 3 �1 1

3 �2

�3�5

3 �2

1 3 �1 1

28 �19 7 �9

2 �5 3 1

5 6 2 1

2 1


Determine whether each matrix product is defined. If so, state the dimensions ofthe product.

1. A7 � 4 � B4 � 3 7 � 3 2. A3 � 5 � M5 � 8 3 � 8 3. M2 � 1 � A1 � 6 2 � 6

4. M3 � 2 � A3 � 2 undefined 5. P1 � 9 � Q9 � 1 1 � 1 6. P9 � 1 � Q1 � 9 9 � 9


7. � 8. �

9. � 10. � not possible

11. [4 0 2] � [2] 12. � [4 0 2]

13. � 14. [�15 �9] � [�297 �75]

Use A � , B � , C � , and scalar c � 3 to determine whether the following equations are true for the given matrices.

15. AC � CA yes 16. A(B � C) � BA � CA no

17. (AB)c � c(AB) yes 18. (A � C)B � B(A � C) no

RENTALS For Exercises 19–21, use the following information.For their one-week vacation, the Montoyas can rent a 2-bedroom condominium for$1796, a 3-bedroom condominium for $2165, or a 4-bedroom condominium for$2538. The table shows the number of units in each of three complexes.

19. Write a matrix that represents the number of each type of unit available at each complex and a matrix that ,represents the weekly charge for each type of unit.

20. If all of the units in the three complexes are rented for the week at the rates given the Montoyas, express the income of each of the three complexes as a matrix.

21. What is the total income of all three complexes for the week? $526,054

172,452227,960125,642

$1796$2165$2538

36 24 2229 32 4218 22 18

2-Bedroom 3-Bedroom 4-Bedroom

Sun Haven 36 24 22

Surfside 29 32 42

Seabreeze 18 22 18

�1 0 0 �1

4 0�2 �1

1 33 1

6 11 23 �10

�30 10 15 �5

5 0 0 5

�6 2 3 �1

4 0 2 12 0 6�4 0 �2

1 3 �1

1 3 �1

3 �2 7 6 0 �5

3 �2 7 6 0 �5

�6 �12 39 3

2 4 7 �1

�3 0 2 5

2 20�23 �5

�3 0 2 5

2 4 7 �1

30 �4 �6 3 �6 26

3 �2 7 6 0 �5

2 4 3 �1

Practice (Average)

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Reading to Learn MathematicsMultiplying Matrices

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Pre-Activity How can matrices be used in sports statistics?


Write a sum that shows the total points scored by the Oakland Raidersduring the 2000 season. (The sum will include multiplications. Do notactually calculate this sum.)

6 � 58 � 1 � 56 � 3 � 23 � 2 � 1 � 2 � 2

Reading the Lesson

1. Determine whether each indicated matrix product is defined. If so, state the dimensionsof the product. If not, write undefined.

a. M3 � 2 and N2 � 3 MN: NM:

b. M1 � 2 and N1 � 2 MN: NM:

c. M4 � 1 and N1 � 4 MN: NM:

d. M3 � 4 and N4 � 4 MN: NM:

2. The regional sales manager for a chain of computer stores wants to compare the revenuefrom sales of one model of notebook computer and one model of printer for three storesin his area. The notebook computer sells for $1850 and the printer for $175. The numberof computers and printers sold at the three stores during September are shown in thefollowing table.

Write a matrix product that the manager could use to find the total revenue forcomputers and printers for each of the three stores. (Do not calculate the product.)

�


3. Many students find the procedure of matrix multiplication confusing at first because itis unfamiliar. Think of an easy way to use the letters R and C to remember how tomultiply matrices and what the dimensions of the product will be. Sample answer:Just remember RC for “row, column.” Multiply each row of the first matrixby each column of the second matrix. The dimensions of the product arethe number of rows of the first matrix and the number of columns of thesecond matrix.

1850 175

128 101205 166 97 73

Store Computers Printers

A 128 101

B 205 166

C 97 73

undefined3 � 4

1 � 14 � 4

undefinedundefined

2 � 23 � 3


Fourth-Order DeterminantsTo find the value of a 4 � 4 determinant, use a method called expansion by minors.

First write the expansion. Use the first row of the determinant.Remember that the signs of the terms alternate.

� � � � � � � � � �Then evaluate each 3 � 3 determinant. Use any row.

� � � � � � � � � �� 2(�16 � 10) � �3(24) � 5(�6)� �52 � �102

� � � � � � � � � �� 6(�16 � 10) � �4(�6) � 3(�12)� �156 � �12

Finally, evaluate the original 4 � 4 determinant.

� �Evaluate each determinant.

1.

� �2.

� �3.

� �1 4 3 0

�2 �3 6 45 1 1 24 2 5 �1

3 3 3 32 1 2 14 3 �1 52 5 0 1

1 2 3 14 3 �1 02 �5 4 41 �2 0 2

� 6(�52) � 3(�102) � 2(�156) � 7(�12) � �846

6 �3 2 70 4 3 50 2 1 �46 0 �2 0

0 26 0� 3

0 16 �2� �4

0 4 30 2 16 0 �2

4 52 �4� 6

0 4 50 2 �46 0 0

0 16 �2� 5

0 �46 0

� �30 3 50 1 �46 �2 0

4 52 �4� �(�2)

4 3 52 1 �40 �2 0

0 4 30 2 16 0 �2

� 70 4 50 2 �46 0 0

� 20 3 50 1 �46 �2 0

� (�3)4 3 52 1 �40 �2 0

� 6

6 �3 2 70 4 3 50 2 1 �46 0 �2 0

Enrichment

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Study Guide and InterventionTransformations with Matrices

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Translations and Dilations Matrices that represent coordinates of points on a planeare useful in describing transformations.

Translation a transformation that moves a figure from one location to another on the coordinate plane

You can use matrix addition and a translation matrix to find the coordinates of thetranslated figure.

Dilation a transformation in which a figure is enlarged or reduced

You can use scalar multiplication to perform dilations.

Find the coordinates of the vertices of the image of �ABC with vertices A(�5, 4), B(�1, 5), and C(�3, �1) if it is moved 6 units to the right and 4 unitsdown. Then graph �ABC and its image �A�B�C�.

Write the vertex matrix for �ABC.

Add the translation matrix to the vertex matrix of �ABC.

� �

The coordinates of the vertices of �A�B�C� are A�(1, 0), B�(5, 1), and C�(3, �5).

For Exercises 1 and 2 use the following information. Quadrilateral QUAD withvertices Q(�1, �3), U(0, 0), A(5, �1), and D(2, �5) is translated 3 units to the leftand 2 units up.

1. Write the translation matrix.

2. Find the coordinates of the vertices of Q�U�A�D�.Q�(�4, �1), U �, (�3, 2), A�(2, 1), D �(�1, �3)

For Exercises 3–5, use the following information. The vertices of �ABC are A(4, �2), B(2, 8), and C(8, 2). The triangle is dilated so that its perimeter is one-fourth the original perimeter.

3. Write the coordinates of the vertices of �ABC in a vertex matrix.

4. Find the coordinates of the vertices of image �A�B�C�.

A��1, � �, B �� , 2�, C ��2, �5. Graph the preimage and the image.

1�2

1�2

1�2

4 2 8�2 8 2

�

�

�x

y

O

�3 �3 �3 �3 2 2 2 2

1 5 3 0 1 �5

6 6 6 �4 �4 �4

�5 �1 �3 4 5 �1

6 6 6 �4 �4 �4

�5 �1 �3 4 5 �1 x

y

OA�

B�

C�

AB

C

ExampleExample

ExercisesExercises


Reflections and Rotations

ReflectionFor a reflection over the: x-axis y-axis line y � x

Matricesmultiply the vertex matrix on the left by:

Rotation

For a counterclockwise rotation about the90° 180° 270°

Matrices

origin of:

multiply the vertex matrix on the left by:

Find the coordinates of the vertices of the image of �ABC with A(3, 5), B(�2, 4), and C(1, �1) after a reflection over the line y � x.Write the ordered pairs as a vertex matrix. Then multiply the vertex matrix by thereflection matrix for y � x.

� �

The coordinates of the vertices of A�B�C� are A�(5, 3), B�(4, �2), and C�(�1, 1).

1. The coordinates of the vertices of quadrilateral ABCD are A(�2, 1), B(�1, 3), C(2, 2), andD(2, �1). What are the coordinates of the vertices of the image A�B�C�D� after areflection over the y-axis? A�(2, 1), B �(1, 3), C �(�2, 2), D �(�2, �1)

2. Triangle DEF with vertices D(�2, 5), E(1, 4), and F(0, �1) is rotated 90°counterclockwise about the origin.

a. Write the coordinates of the triangle in a vertex matrix.

b. Write the rotation matrix for this situation.

c. Find the coordinates of the vertices of �D�E�F�.D �(�5, �2), E �(�4, 1), F �(1, 0)

d. Graph �DEF and �D�E�F�.

�

�

� x

y

O

0 �11 0

�2 1 0 5 4 �1

5 4 �1 3 �2 1

3 �2 1 5 4 �1

0 1 1 0

0 1 �1 0

�1 0 0 �1

0 �1 1 0

0 1 1 0

�1 0 0 1

1 0 0 �1


Transformations with Matrices

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For Exercises 1–3, use the following information.Triangle ABC with vertices A(2, 3), B(0, 4), and C(�3, �3) is translated 3 units right and 1 unit down.


2. Find the coordinates of �A�B�C�. A�(5, 2), B�(3, 3), C �(0, �4)3. Graph the preimage and the image.

For Exercises 4–6, use the following information.The vertices of �RST are R(�3, 1), S(2, �1), and T(1, 3). The triangle is dilated so that its perimeter is twice the original perimeter.

4. Write the coordinates of �RST in a vertex matrix.

5. Find the coordinates of the image �R�S�T�.R �(�6, 2), S �(4, �2), T �(2, 6)

6. Graph �RST and �R�S�T�.

For Exercises 7–10, use the following information.The vertices of �DEF are D(4, 0), E(0, �1), and F(2, 3).The triangle is reflected over the x-axis.

7. Write the coordinates of �DEF in a vertex matrix.

8. Write the reflection matrix for this situation.

9. Find the coordinates of �D�E�F �. D �(4, 0), E �(0, 1), F �(2, �3)

10. Graph �DEF and �D�E�F �.

For Exercises 11–14, use the following information.Triangle XYZ with vertices X(1, �3), Y(�4, 1), and Z(�2, 5) is rotated 180º counterclockwise about the origin.

11. Write the coordinates of the triangle in a vertex matrix.

12. Write the rotation matrix for this situation.

13. Find the coordinates of �X�Y�Z�.X �(�1, 3), Y �(4, �1), Z�(2, �5)

14. Graph the preimage and the image.

�1 0 0 �1

1 �4 �2�3 1 5

�

�

�

x

y

O

1 00 �1 �

�

�

x

y

O

4 0 20 �1 3

�3 2 1 1 �1 3 �

�

�

x

y

O

3 3 3�1 �1 �1

�

�

�

x

y

O


For Exercises 1–3, use the following information.Quadrilateral WXYZ with vertices W(�3, 2), X(�2, 4), Y(4, 1), and Z(3, 0) is translated 1 unit left and 3 units down.


2. Find the coordinates of quadrilateral W�X�Y�Z�.W�(�4, �1), X �(�3, 1), Y �(3, �2), Z �(2, �3)

3. Graph the preimage and the image.

For Exercises 4–6, use the following information.The vertices of �RST are R(6, 2), S(3, �3), and T(�2, 5). The triangle is dilated so that its perimeter is one half the original perimeter.

4. Write the coordinates of �RST in a vertex matrix.

5. Find the coordinates of the image �R�S�T�.R �(3, 1), S �(1.5, �1.5), T �(�1, 2.5)

6. Graph �RST and �R�S�T�.

For Exercises 7–10, use the following information.The vertices of quadrilateral ABCD are A(�3, 2), B(0, 3), C(4, �4),and D(�2, �2). The quadrilateral is reflected over the y-axis.

7. Write the coordinates of ABCD in a vertex matrix.

8. Write the reflection matrix for this situation.

9. Find the coordinates of A�B�C�D�.A�(3, 2), B �(0, 3), C �(�4, �4), D �(2, �2)

10. Graph ABCD and A�B�C�D�.

11. ARCHITECTURE Using architectural design software, the Bradleys plot their kitchenplans on a grid with each unit representing 1 foot. They place the corners of an island at(2, 8), (8, 11), (3, 5), and (9, 8). If the Bradleys wish to move the island 1.5 feet to theright and 2 feet down, what will the new coordinates of its corners be?(3.5, 6), (9.5, 9), (4.5, 3), and (10.5, 6)

12. BUSINESS The design of a business logo calls for locating the vertices of a triangle at(1.5, 5), (4, 1), and (1, 0) on a grid. If design changes require rotating the triangle 90ºcounterclockwise, what will the new coordinates of the vertices be?(�5, 1.5), (�1, 4), and (0, 1)

�1 0 0 1

��

�

�

x

y

O

�3 0 4 �2 2 3 �4 �2

6 3 �22 �3 5

�

�

x

y

O

�1 �1 �1 �1�3 �3 �3 �3

�

�

�

�

x

y

O

Practice (Average)

Transformations with Matrices

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Reading to Learn MathematicsTransformations with Matrices

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Pre-Activity How are transformations used in computer animation?


Describe how you can change the orientation of a figure without changingits size or shape.

Flip (or reflect) the figure over a line.

Reading the Lesson

1. a. Write the vertex matrix for the quadrilateral ABCD shown in the graph at the right.

b. Write the vertex matrix that represents the position of the quadrilateral A�B�C�D� thatresults when quadrilateral ABCD is translated 3 units to the right and 2 units down.

2. Describe the transformation that corresponds to each of the following matrices.

a. b.

counterclockwise rotation translation 4 units down and about the origin of 180� 3 units to the right

c. d.

reflection over the y-axis reflection over the line y � x


3. Describe a way to remember which of the reflection matrices corresponds to reflectionover the x-axis.

Sample answer: The only elements used in the reflection matrices are 0,1, and �1. For such a 2 � 2 matrix M to have the property that

M � � , the elements in the top row must be 1 and 0 (in that

order), and elements in the bottom row must be 0 and �1 (in that order).

x�y

xy

0 1 1 0

�1 0 0 1

3 3 3 �4 �4 �4

�1 0 0 �1

�1 5 4 1�1 1 �6 �5

�4 2 1 �2 1 3 �4 �3 x

y

O

A

B

CD


Properties of DeterminantsThe following properties often help when evaluating determinants.

• If all the elements of a row (or column) are zero, the value of the determinant is zero.

� 0

• Multiplying all the elements of a row (or column) by a constant is equivalent to multiplying the value of the determinant by the constant.

3 �

• If two rows (or columns) have equal corresponding elements, the value of thedeterminant is zero.

� 0

• The value of a determinant is unchanged if any multiple of a row (or column) is added to corresponding elements of another row (or column).

�

(Row 2 is added to row 1.)

• If two rows (or columns) are interchanged, the sign of the determinant is changed.

�

• The value of the determinant is unchanged if row 1 is interchanged with column 1,and row 2 is interchanged with column 2. The result is called the transpose.

�

Exercises 1–6

Verify each property above by evaluating the given determinants and give another example of the property.

5 3 �7 4

5 �7 3 4

�3 8 5 5

4 5 �3 8

6 2 2 5

4 �3 2 5

5 5 �3 �3

12 �3 5 3

4 �1 5 3

a b0 0

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Determinants of 2 � 2 Matrices

Second-Order Determinant For the matrix , the determinant is � ad � bc.

Find the value of each determinant.

a.

� 6(5) � 3(�8)

� 30 �(�24) or 54

b.

� 11(�3) � (�5)(9)

� �33 � (�45) or 12


1. 52 2. �2 3. �18

4. 64 5. �6 6. 1

7. �114 8. 144 9. �335

10. 48 11. �16 12. 495

13. 15 14. 36 15. 17

16. 18.3 17. 18. 37 6.8 15 �0.2 5

13�60

�23� ��

12�

�16� �

15�

4.8 2.1 3.4 5.3

20 110 0.1 1.4

8.6 0.5 14 5

0.2 8 �1.5 15

13 62 �4 19

24 �8 7 �3

10 16 22 40

�8 35 5 20

15 12 23 28

14 8 9 �3

4 7 5 9

�6 �3 �4 �1

5 12 �7 �4

3 9 4 6

�8 3 �2 1

6 �2 5 7

11 �5 9 3

11 �5 9 3

6 3 �8 5

6 3�8 5

a b c d

a b c d

ExampleExample

ExercisesExercises


Determinants of 3 � 3 Matrices

Third-Order Determinants � a � b � c

Area of a Triangle

The area of a triangle having vertices (a, b), (c, d ) and (e, f ) is |A |, where

A � .

Evaluate .

� 4 � 5 � 2 Third-order determinant

� 4(18 � 0) � 5(6 � 0) � 2(�3 � 6) Evaluate 2 � 2 determinants.

� 4(18) � 5(6) � 2(�9) Simplify.

� 72 � 30 � 18 Multiply.

� 60 Simplify.

Evaluate each determinant.

1. �57 2. 80 3. 28

4. 54 5. �63 6. �2

7. Find the area of a triangle with vertices X(2, �3), Y(7, 4), and Z(�5, 5).44.5 square units

5 �4 1 2 3 �2 �1 6 �3

6 1 �4 3 2 1 �2 2 �1

5 �2 2 3 0 �2 2 4 �3

6 1 4 �2 3 0 �1 3 2

4 1 0 �2 3 1 2 �2 5

3 �2 �2 0 4 1 �1 5 �3

1 3 2 �3

1 0 2 6

3 0 �3 6

4 5 �2 1 3 0 2 �3 6

4 5 �21 3 02 �3 6

a b 1 c d 1 e f 1

1�2

d e g h

d f g i

e f h i

a b c d e f g h i


Determinants

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1. 13 2. 35 3. 1

4. �13 5. �45 6. 0

7. 14 8. 13 9. 1

10. 11 11. �5 12. �52

13. �3 14. 17 15. 68

16. �4 17. 10 18. �17

Evaluate each determinant using expansion by minors.

19. �1 20. �2 21. �1

Evaluate each determinant using diagonals.

22. �3 23. 8 24. 40 3 2 2 1 �1 4 3 �1 0

3 �1 2 1 0 4 3 �2 0

2 �1 6 3 2 5 2 3 1

2 6 1 3 5 �1 2 1 �2

6 �1 1 5 2 �1 1 3 �2

2 �1 1 3 2 �1 2 3 �2

�1 6 2 5

2 2 �1 4

�1 2 0 4

�1 �14 5 2

�1 �3 5 �2

3 �5 6 �11

�12 4 1 4

1 �3 �3 4

1 �5 1 6

9 �2 �4 1

�3 1 8 �7

�5 2 8 �6

3 12 2 8

0 9 5 8

2 5 3 1

1 6 1 7

10 9 5 8

5 2 1 3



1. �5 2. 0 3. �18

4. 34 5. �20 6. 3

7. 36 8. 55 9. 112

10. 30 11. �16 12. 0.13

Evaluate each determinant using expansion by minors.

13. �48 14. 45 15. 7

16. 28 17. �72 18. 318

Evaluate each determinant using diagonals.

19. 10 20. 12 21. 5

22. 20 23. �4 24. 0

25. GEOMETRY Find the area of a triangle whose vertices have coordinates (3, 5), (6, �5),and (�4, 10). 27.5 units2

26. LAND MANAGEMENT A fish and wildlife management organization uses a GIS(geographic information system) to store and analyze data for the parcels of land itmanages. All of the parcels are mapped on a grid in which 1 unit represents 1 acre. Ifthe coordinates of the corners of a parcel are (�8, 10), (6, 17), and (2, �4), how manyacres is the parcel? 133 acres

2 1 3 1 8 0 0 5 �1

2 �1 �2 4 0 �2 0 3 2

1 2 �4 1 4 �6 2 3 3

1 �4 �1 1 �6 �2 2 3 1

2 2 3 1 �1 1 3 1 1

�4 3 �1 2 1 �2 4 1 �4

�12 0 3 7 5 �1 4 2 �6

2 7 �6 8 4 0 1 �1 3

0 �4 0 2 �1 1 3 �2 5

2 1 1 1 �1 �2 1 1 �1

2 �4 1 3 0 9 �1 5 7

�2 3 1 0 4 �3 2 5 �1

0.5 �0.7 0.4 �0.3

2 �13 �9.5

3 �4 3.75 5

�1 �11 10 �2

3 �4 7 9

4 0 �2 9

2 �5 5 �11

4 �3 �12 4

�14 �3 2 �2

4 1 �2 �5

9 6 3 2

1 6 2 7

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Determinants

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Pre-Activity How are determinants used to find areas of polygons?


In this lesson, you will learn how to find the area of a triangle if you know thecoordinates of its vertices using determinants. Describe a method you alreadyknow for finding the area of the Bermuda Triangle. Sample answer:Use the map. Choose any side of the triangle as the base, andmeasure this side with a ruler. Multiply this length by the scalefactor for the map. Next, draw a segment from the oppositevertex perpendicular to the base. Measure this segment, andmultiply its length by the scale for the map. Finally, find the area by using the formula A � bh.

Reading the Lesson

1. Indicate whether each of the following statements is true or false.

a. Every matrix has a determinant. false

b. If both rows of a 2 � 2 matrix are identical, the determinant of the matrix will be 0. true

c. Every element of a 3 � 3 matrix has a minor. true

d. In order to evaluate a third-order determinant by expansion by minors it is necessaryto find the minor of every element of the matrix. false

e. If you evaluate a third-order determinant by expansion about the second row, theposition signs you will use are � � �. true

2. Suppose that triangle RST has vertices R(�2, 5), S(4, 1), and T(0, 6).

a. Write a determinant that could be used in finding the area of triangle RST.

b. Explain how you would use the determinant you wrote in part a to find the area ofthe triangle. Sample answer: Evaluate the determinant and multiply the result by . Then take the absolute value to make sure the final answer

is positive.


3. A good way to remember a complicated procedure is to break it down into steps. Write alist of steps for evaluating a third-order determinant using expansion by minors.Sample answer: 1. Choose a row of the matrix. 2. Find the position signsfor the row you have chosen. 3. Find the minor of each element in thatrow. 4. Multiply each element by its position sign and by its minor. 5. Addthe results.

1�2

�2 5 1 4 1 1 0 6 1

1�2


Matrix MultiplicationA furniture manufacturer makes upholstered chairs and wood tables. Matrix Ashows the number of hours spent on each item by three different workers. One day the factory receives an order for 10 chairs and 3 tables. This is shown in matrix B.

� A � B

[10 3] � [10(4) � 3(18) 10(2) � 3(15) 10(12) � 3(0)] � [94 65 120]

The product of the two matrices shows the number of hours needed for each type of worker to complete the order: 94 hours for woodworking, 65 hours for finishing, and 120 hours for upholstering.

To find the total labor cost, multiply by a matrix that shows the hourly rate for each worker: $15 for woodworking, $9 for finishing, and $12 for upholstering.

C � [94 65 120] � [94(15) � 65(9) � 120(12)] � $3435

Use matrix multiplication to solve these problems.A candy company packages caramels, chocolates, and hard candy in three different assortments: traditional, deluxe, and superb. For each type of candy the table below gives the number in each assortment, the number of Calories per piece, and the cost to make each piece.

Calories cost per traditional deluxe superb per piece piece (cents)

caramels 10 16 15 60 10chocolates 12 8 25 70 12card candy 10 16 8 55 6

The company receives an order for 300 traditional, 180 deluxe and100 superb assortments.

1. Find the number of each type of candy needed to fill the order.7380 caramels; 7540 chocolates; 6680 hard candies

2. Find the total number of Calories in each type of assortment.1990-traditional; 2400-deluxe; 3090-superb

3. Find the cost of production for each type of assortment.$3.04-traditional; $3.52-deluxe; $4.98-superb

4. Find the cost to fill the order. $2043.60

15 9 12

4 2 12 18 15 0

chair tablenumber ordered [ 10 3 ]

hourswoodworker finsher upholsterer

chair 4 2 12 table 18 15 0

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Systems of Two Linear Equations Determinants provide a way for solving systemsof equations.

Cramer’s Rule for

The solution of the linear system of equations ax � by � e

Two-Variable Systems

cx � dy � f

is (x, y ) where x � , y � , and � 0.

Use Cramer’s Rule to solve the system of equations. 5x � 10y � 810x � 25y � �2

x � Cramer’s Rule y �

� a � 5, b � �10, c � 10, d � 25, e � 8, f � �2 �

� Evaluate each determinant. �

� or Simplify. � � or �

The solution is � , � �.

Use Cramer’s Rule to solve each system of equations.

1. 3x � 2y � 7 2. x � 4y � 17 3. 2x � y � �22x � 7y � 38 (5, 4) 3x � y � 29 (9, �2) 4x � y � 4 (3, 8)

4. 2x � y � 1 5. 4x � 2y � 1 6. 6x � 3y � �35x � 2y � �29 (�3, �7) 5x � 4y � 24 �2, � � 2x � y � 21 (5, 11)

7. 2x � 7y � 16 8. 2x � 3y � �2 9. � � 2

x � 2y � 30 (22, �4) 3x � 4y � 9 (35, 24) � � �8 (�12, 30)

10. 6x � 9y � �1 11. 3x � 12y � �14 12. 8x � 2y �

3x � 18y � 12 9x � 6y � �75x � 4y � �

� , � �� , � �� , �11�14

1�7

5�6

4�3

5�9

2�3

27�7

3�7

y�6

x�4

y�5

x�3

7�2

2�5

4�5

2�5

90�225

4�5

180�225

5(�2) � 8(10)��5(25) � (�10)(10)

8(25) � (�2)(�10)��5(25) � (�10)(10)

5 8 10 �2�� 5 �10 10 25

8 �10 �2 25�� 5 �10 10 25

a e c f � a b c d

e b f d � a b c d

a b c d

a e c f � a b c d

e b f d � a b c d

ExampleExample

ExercisesExercises


Systems of Three Linear Equations

The solution of the system whose equations areax � by � cz � j

Cramer’s Rule for

dx � ey � fz � k

Three-Variable Systems

gx � hy � iz � l

is (x, y, z) where x � , y � , and z � and � 0.

Use Cramer’s rule to solve the system of equations.6x � 4y � z � 52x � 3y �2z � �28x � 2y � 2z � 10Use the coefficients and constants from the equations to form the determinants. Thenevaluate each determinant.

x � y � z �

� or � or � � or

The solution is � , � , �.

Use Cramer’s rule to solve each system of equations.

1. x � 2y � 3z � 6 2. 3x � y � 2z � �22x � y � z � �3 4x � 2y � 5z � 7x � y � z � 6 (1, 2, 3) x � y � z � 1 (3, �5, 3)

3. x � 3y � z � 1 4. 2x � y � 3z � �52x � 2y � z � �8 x � y � 5z � 214x � 7y � 2z � 11 (�2, 1, 6) 3x � 2y � 4z � 6 (4, 7, �2)

5. 3x � y � 4z � 7 6. 2x � y � 4z � 92x � y � 5z � �24 3x � 2y � 5z � �1310x � 3y � 2z � �2 (�3, 8, �2) x � y � 7z � 0 (5, 9, 2)

4�3

1�3

5�6

4�3

�128��96

1�3

32��96

5�6

�80��96

6 4 5 2 3 �2 8 �2 10�� 6 4 1 2 3 �2 8 �2 2

6 5 1 2 �2 �2 8 10 2�� 6 4 1 2 3 �2 8 �2 2

5 4 1 �2 3 �2 10 �2 2�� 6 4 1 2 3 �2 8 �2 2

a b c d e f g h i

a b j d e k g h l � a b c d e f g h i

a j c d k f g l i � a b c d e f g h i

j b c k e f l h i � a b c d e f g h i


Cramer’s Rule

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1. 2a � 3b � 6 2. 3x � y � 22a � b � �2 (�3, 4) 2x � y � 3 (1, �1)

3. 2m � 3n � �6 4. x � y � 2m � 3n � 6 (0, �2) 2x � 3y � 9 (3, 1)

5. 2x � y � 4 6. 3r � s � 77x � 2y � 3 (1, 2) 5r � 2s � 8 (6, 11)

7. 4g � 5h � 1 8. 7x � 5y � �8g � 3h � 2 (�1, 1) 9x � 2y � 3 (1, �3)

9. 3x � 4y � 2 10. 2x � y � 54x � 3y � 12 (6, 4) 3x � y � 5 (2, �1)

11. 3p � 6q � 18 12. x � 2y � �12p � 3q � 5 (4, �1) 2x � y � 3 (1, 1)

13. 5c � 3d � 5 14. 5t � 2v � 22c � 9d � 2 (1, 0) 2t � 3v � �8 (2, �4)

15. 5a � 2b � 14 16. 65w � 8z � 833a � 4b � 11 (3, 0.5) 9w � 4z � 0 (1, �2.25)

17. GEOMETRY The two sides of an angle are contained in the lines whose equations are3x � 2y � 4 and x � 3y � 5. Find the coordinates of the vertex of the angle. (2, �1)


18. a � b � 5c � 2 19. x � 3y � z � 53a � b � 2c � 3 2x � 5y � z � 124a � 2b � c � �3 (2, �5, 1) x � 2y � 3z � �13 ( 3, 2, 4)

20. 3c � 5d � 2e � 4 21. r � 4s � t � 62c � 3d � 4c � �3 2r � s � 3t � 04c � 2d � 3e � 0 (1, �1, �2) 3r � 2s � t � 4 (1, �1, �1)



1. 2x � y � 0 2. 5c � 9d � 19 3. 2x � 3y � 53x � 2y � �2 (2, �4) 2c � d � �20 (�7, 6) 3x � 2y � 1 (1, 1)

4. 20m � 3n � 28 5. x � 3y � 6 6. 5x � 6y � �452m � 3n � 16 (2, 4) 3x � y � �22 (�6, �4) 9x � 8y � 13 (�3, 5)

7. �2e � f � 4 8. 2x � y � �1 9. 8a � 3b � 24�3e � 5f � �15 (�5, �6) 2x � 4y � 8 (�2, �3) 2a � b � 4 (6, �8)

10. �3x � 15y � 45 11. 3u � 5v � 11 12. �6g � h � �10�2x � 7y � 18 (5, 4) 6u � 7v � �12 � , �2� �3g � 4h � 4 � , �2�

13. x � 3y � 8 14. 0.2x � 0.5y � �1 15. 0.3d � 0.6g � 1.8x � 0.5y � 3 (2, �2) 0.6x � 3y � �9 (5, 4) 0.2d � 0.3g � 0.5 (4, �1)

16. GEOMETRY The two sides of an angle are contained in the lines whose equations are

x � y � 6 and 2x � y � 1. Find the coordinates of the vertex of the angle. (2, �3)

17. GEOMETRY Two sides of a parallelogram are contained in the lines whose equationsare 0.2x � 0.5y � 1 and 0.02x � 0.3y � �0.9. Find the coordinates of a vertex of theparallelogram. (15, 4)


18. x � 3y � 3z � 4 19. �5a � b � 4c � 7 20. 2x � y � 3z � �5�x � 2y � z � �1 �3a � 2b � c � 0 5x � 2y � 2z � 84x � y � 2z � �1 2a � 3b � c � 17 3x � 3y � 5z � 17

(1, �1, 2) (3, 2, �5) (2, 3, 4)

21. 2c � 3d � e � 17 22. 2j � k � 3m � �3 23. 3x � 2y � 5z � 34c � d � 5e � �9 3j � 2k � 4m � 5 2x � 2y � 4z � 3c � 2d � e � 12 �4j � k � 2m � 4 �5x � 10y � 7z � �3

(2, 3, �4) (�1, 2, 1) (1.2, 0.3, 0)

24. LANDSCAPING A memorial garden being planted in front of a municipal library will contain three circular beds that are tangent to each other. A landscape architect has prepared a sketch of the design for the garden using CAD (computer-aided drafting) software, as shown at theright. The centers of the three circular beds are representedby points A, B, and C. The distance from A to B is 15 feet,the distance from B to C is 13 feet, and the distance from A to C is 16 feet. What is the radius of each of the circularbeds? circle A: 9 ft, circle B: 6 ft, circle C: 7 ft

A

B C

13 ft

16 ft15 ft

4�3

4�3

1�3

Practice (Average)

Cramer’s Rule

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Reading to Learn MathematicsCramer’s Rule

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Pre-Activity How is Cramer’s Rule used to solve systems of equations?


A triangle is bounded by the x-axis, the line y � x, and the line

y � �2x � 10. Write three systems of equations that you could use to findthe three vertices of the triangle. (Do not actually find the vertices.)

x � 0, y � x; x � 0, y � �2x � 10; y � x, y � �2x � 10

Reading the Lesson

1. Suppose that you are asked to solve the following system of equations by Cramer’s Rule.

3x � 2y � 72x � 3y � 22

Without actually evaluating any determinants, indicate which of the following ratios ofdeterminants gives the correct value for x. B

A. B. C.

2. In your textbook, the statements of Cramer’s Rule for two variables and three variablesspecify that the determinant formed from the coefficients of the variables cannot be 0.If the determinant is zero, what do you know about the system and its solutions?

The system could be a dependent system and have infinitely manysolutions, or it could be an inconsistent system and have no solutions.


3. Some students have trouble remembering how to arrange the determinants that are usedin solving a system of two linear equations by Cramer’s Rule. What is a good way toremember this?

Sample answer: Let D be the determinant of the coefficients. Let Dx be the determinant formed by replacing the first column of D with the constantsfrom the right-hand side of the system, and let Dy be the determinantformed by replacing the second column of D with the constants. Then

x � and y � .Dy�D

Dx�D

3 7 2 22�� 3 2 2 �3

7 2 22 �3�� 3 2 2 �3

3 2 2 �3�� 7 2 22 �3

1�2

1�2

1�2


Communications NetworksThe diagram at the right represents a communications network linking five computer remote stations. The arrows indicate thedirection in which signals can be transmitted and received by eachcomputer. We can generate a matrix to describe this network.

A �

Matrix A is a communications network for direct communication. Suppose you want to send a message from one computer to another using exactly one other computer as a relay point. It can be shown that the entries of matrix A2 represent the number of ways to send a message from one point to another by going through a third station. For example, a message may be sent from station 1 to station 5 by going through station 2 or station 4 on the way. Therefore, A2

1,5 � 2.

A2 �

For each network, find the matrices A and A2. Then write the number of ways the messages can be sent for each matrix.

1. 2. 3.

A: A: A:

9 ways11 ways 10 ways

A: A: A:

21 ways21 ways 14 ways

1 0 0 0 00 1 0 0 01 1 1 0 01 1 0 1 03 1 1 1 0

1 0 1 0 1 00 1 0 0 1 11 0 2 1 0 00 1 1 2 0 01 1 0 0 2 10 1 0 1 0 0

3 1 2 01 2 1 11 1 2 11 2 1 1

0 1 0 0 01 0 0 0 01 0 0 1 01 0 1 0 01 1 1 1 0

0 1 0 1 0 00 0 1 0 0 00 1 0 0 1 11 0 0 0 1 00 0 1 1 0 01 0 0 0 0 0

0 1 1 11 0 1 01 1 0 01 0 1 0

2

5

1

3 4

2

6

1 3

4 5

1

2 3 4

Again, compare the entries of matrix A2 to thecommunications diagram to verify that the entries arecorrect. Matrix A2 represents using exactly one relay.

to computer j

1 1 1 1 2from

1 2 1 1 1computer i 1 2 1 1 1

1 2 0 4 1 1 1 1 1 2

The entry in position aij represents the number of ways to senda message from computer i to computer j directly. Compare theentries of matrix A to the diagram to verify the entries. Forexample, there is one way to send a message from computer 3to computer 4, so A3,4 � 1. A computer cannot send a messageto itself, so A1,1 � 0.

to computer j

0 1 0 1 0from

0 0 0 1 1computer i 1 0 0 1 0

1 1 1 0 1 0 1 0 1 0

1 2

3 4

5

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Identity and Inverse Matrices The identity matrix for matrix multiplication is asquare matrix with 1s for every element of the main diagonal and zeros elsewhere.

Identity Matrix If A is an n � n matrix and I is the identity matrix,for Multiplication then A � I � A and I � A � A.

If an n � n matrix A has an inverse A�1, then A � A�1 � A�1 � A � I.

Determine whether X � and Y � are inversematrices.Find X � Y.

X � Y � �

� or

Find Y � X.

Y � X � �

� or

Since X � Y � Y � X � I, X and Y are inverse matrices.

Determine whether each pair of matrices are inverses.

1. and 2. and 3. and

yes yes no


yes no yes


no yes no

10. and 11. and 12. and

no yes yes

�5 3 7 �4

4 3 7 5

5 �2 �17 7

7 2 17 5

3 2 �4 �3

3 2 4 �6

�72� ��

32�

1 �2 3 7 2 4

�3 4 2 ��

52�

5 8 4 6

��15� ��1

10�

�1

30� �1

10�

4 2 6 �2

�2 1 �

121� ��

52�

5 2 11 4

1 2 3 8

4 �1 5 3

�4 11 3 �8

8 11 3 4

2 3 �1 �2

2 3 5 �1

2 �1 ��

52� �

32�

3 2 5 4

4 �5 �3 4

4 5 3 4

1 0 0 1

21 � 20 12 � 12 �35 � 35 �20 � 21

7 4 10 6

3 �2 �5 �

72�

1 0 0 1

21 � 20 �14 � 14 30 � 30 �20 � 21

3 �2 �5 �

72�

7 4 10 6

3 �2 �5 �

72�

7 410 6

ExampleExample

ExercisesExercises


Find Inverse Matrices

Inverse of a 2 � 2 MatrixThe inverse of a matrix A � is

A�1 � , where ad � bc � 0.

If ad � bc � 0, the matrix does not have an inverse.

Find the inverse of N � .

First find the value of the determinant.

� 7 � 4 � 3

Since the determinant does not equal 0, N�1 exists.

N�1 � � �

Check:

NN�1 � � � �

N�1N � � � �

Find the inverse of each matrix, if it exists.

1. 2. 3.

no inverse exists

4. 5. 6.

��12

� no inverse exists 4 �2�10 �8

1�12

8 �5�10 6

8 2 10 4

3 6 4 8

6 5 10 8

30 1020 40

1�1000

1 �10 1

40 �10 �20 30

1 1 0 1

24 12 8 4

1 0 0 1

�73� � �

43� �

23� � �

23�

��

134� � �

134� ��

43� � �

73�

7 2 2 1

�13� ��

23�

��

23� �

73�

1 0 0 1

�73� � �

43� ��

134� � �

134�

�

23� � �

23� ��

43� � �

73�

�13� ��

23�

��

23� �

73�

7 2 2 1

�13� ��

23�

��

23� �

73�

1 �2 �2 7

1�3

d �b �c a

1�ad � bc

7 2 2 1

7 22 1

d �b �c a

1�ad � bc

a b c d


Identity and Inverse Matrices

NAME ______________________________________________ DATE ____________ PERIOD _____

4-74-7

ExampleExample

ExercisesExercises

Skills PracticeIdentity and Inverse Matrices

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4-74-7


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1. X � , Y � yes 2. P � , Q � yes

3. M � , N � no 4. A � , B � yes

5. V � , W � yes 6. X � , Y � yes

7. G � , H � yes 8. D � , E � no


9. � 10. �

11. no inverse exists 12.

13. 14. no inverse exists

15. 16. �

17. � 18. no inverse exists

19. � 20. 2 00 2

1�4

2 0 0 2

�8 �8�10 10

1�160

10 8 10 �8

10 8 5 4

0 77 0

1�49

0 �7 �7 0

2 �5�1 �4

1�13

�4 5 1 2

�1 1�1 �1

1�2

�1 �1 1 �1

3 6 �1 �2

3 1�3 1

1�6

1 �1 3 3

0 4�6 �2

1�24

�2 �4 6 0

9 3 6 2

2 �1�3 1

1 1 3 2

0 �2�4 0

1�8

0 2 4 0

�0.125 �0.125 �0.125 �0.125

�4 �4 �4 4

�121� �1

31�

��1

11� �1

41�

4 �3 1 2

��13� �

23�

�

16� �

16�

�1 4 1 2

0 ��17�

�

17� 0

0 7 �7 0

2 �5 1 �2

�2 5 �1 2

�1 0 0 �3

�1 0 0 3

�1 3 1 �2

2 3 1 1

1 0 �1 1

1 0 1 1



1. M � , N � no 2. X � , Y � yes

3. A � , B � yes 4. P � , Q � yes

Determine whether each statement is true or false.

5. All square matrices have multiplicative inverses. false

6. All square matrices have multiplicative identities. true


7. 8.

9. � 10.

11. 12. no inverse exists

GEOMETRY For Exercises 13–16, use the figure at the right.

13. Write the vertex matrix A for the rectangle.

14. Use matrix multiplication to find BA if B � .

15. Graph the vertices of the transformed triangle on the previous graph.Describe the transformation. dilation by a scale factor of 1.5

16. Make a conjecture about what transformation B�1 describes on a coordinate plane.

dilation by a scale factor of

17. CODES Use the alphabet table below and the inverse of coding matrix C � todecode this message:19 | 14 | 11 | 13 | 11 | 22 | 55 | 65 | 57 | 60 | 2 | 1 | 52 | 47 | 33 | 51 | 56 | 55.

CHECK_YOUR_ANSWERSCODE

A 1 B 2 C 3 D 4 E 5 F 6 G 7

H 8 I 9 J 10 K 11 L 12 M 13 N 14

O 15 P 16 Q 17 R 18 S 19 T 20 U 21

V 22 W 23 X 24 Y 25 Z 26 – 0

1 2 2 1

2�3

1.5 6 7.5 3 3 6 1.5 �1.5

1.5 0 0 1.5

1 4 5 22 4 1 �1

x

y

O

(2, –1)

(1, 2)

(4, 4)

(5, 1)

4 6 6 9 1 5

�3 21

�17

2 �5 3 1

3 �51 2

1�11

2 5 �1 3�7 �3

�4 �11�5

�1 3 4 �7

5 0�3 2

1�10

2 0 3 5�3 �5

4 41�8

4 5 �4 �3

�134� �

17�

�

17� �

37�

6 �2 �2 3

�15� ��1

10�

�

25� �1

30�

3 1 �4 2

3 2 5 3

�3 2 5 �3

�2 1 3 �2

2 1 3 2

Practice (Average)

Identity and Inverse Matrices

NAME ______________________________________________ DATE ____________ PERIOD _____

4-74-7

Reading to Learn MathematicsIdentity and Inverse Matrices

NAME ______________________________________________ DATE ____________ PERIOD _____

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Less

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4-7

Pre-Activity How are inverse matrices used in cryptography?


Refer to the code table given in the introduction to this lesson. Suppose thatyou receive a message coded by this system as follows:

16 12 5 1 19 5 2 5 13 25 6 18 9 5 14 4.

Decode the message. Please be my friend.

Reading the Lesson

1. Indicate whether each of the following statements is true or false.

a. Every element of an identity matrix is 1. false

b. There is a 3 � 2 identity matrix. false

c. Two matrices are inverses of each other if their product is the identity matrix. true

d. If M is a matrix, M�1 represents the reciprocal of M. false

e. No 3 � 2 matrix has an inverse. true

f. Every square matrix has an inverse. false

g. If the two columns of a 2 � 2 matrix are identical, the matrix does not have aninverse. true

2. Explain how to find the inverse of a 2 � 2 matrix. Do not use any special mathematicalsymbols in your explanation.

Sample answer: First find the determinant of the matrix. If it is zero, thenthe matrix has no inverse. If the determinant is not zero, form a newmatrix as follows. Interchange the top left and bottom right elements.Change the signs but not the positions of the other two elements.Multiply the resulting matrix by the reciprocal of the determinant of theoriginal matrix. The resulting matrix is the inverse of the original matrix.


3. One way to remember something is to explain it to another person. Suppose that you arestudying with a classmate who is having trouble remembering how to find the inverse ofa 2 � 2 matrix. He remembers how to move elements and change signs in the matrix,but thinks that he should multiply by the determinant of the original matrix. How canyou help him remember that he must multiply by the reciprocal of this determinant?

Sample answer: If the determinant of the matrix is 0, its reciprocal isundefined. This agrees with the fact that if the determinant of a matrix is0, the matrix does not have an inverse.


Permutation MatricesA permutation matrix is a square matrix in which each row and each column has one entry that is 1.All the other entries are 0. Find the inverse of apermutation matrix interchanging the rows andcolumns. For example, row 1 is interchanged withcolumn 1, row 2 is interchanged with column 2.

P is a 4 � 4 permutation matrix. P�1 is the inverse of P.

Solve each problem.

1. There is just one 2 � 2 permutation 2. Find the inverse of the matrix you wrote matrix that is not also an identity in Exercise 1. What do you notice?matrix. Write this matrix.

The two matricesare the same.

3. Show that the two matrices in Exercises 1 and 2 are inverses.

4. Write the inverse of this matrix.

B � B 1

5. Use B�1 from problem 4. Verify that B and B�1 are inverses.

6. Permutation matrices can be used to write and decipher codes. To see how this is done, use the message matrix M and matrix B from problem 4. Find matrix C so that C equals the product MB. Use the rules below.

0 times a letter � 0

1 times a letter � the same letter

0 plus a letter � the same letter

7. Now find the product CB�1. What do you notice?

S H E S A W H I M

0 1 00 0 1

H E S A W S I M H

0 1 00 0 11 0 0

0 � 0 � 0 � 0 � 1 � 1 0 � 1 � 0 � 0 � 1 � 0 0 � 0 � 0 � 1 � 1 � 01 � 0 � 0 � 0 � 0 � 1 1 � 1 � 0 � 0 � 0 � 0 1 � 0 � 0 � 1 � 0 � 00 � 0 � 1 � 0 � 0 � 1 0 � 1 � 1 � 0 � 0 � 0 0 � 0 � 1 � 1 � 0 � 0

0 1 00 0 11 0 0

0 0 1 1 0 0 0 1 0

1 00 1

0 � 1 � 1 � 1 0 � 1 � 1 � 01 � 0 � 0 � 1 1 � 1 � 0 � 0

0 11 0

0 11 0

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

4-74-7

P � P�1 �

0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 0

0 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0

M � H E S A W S I M H

S H E S A W H I M

Study Guide and InterventionUsing Matrices to Solve Systems of Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

4-84-8


Less

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Write Matrix Equations A matrix equation for a system of equations consists of theproduct of the coefficient and variable matrices on the left and the constant matrix on theright of the equals sign.

Write a matrix equation for each system of equations.

a. 3x � 7y � 12x � 5y � �8Determine the coefficient, variable, andconstant matrices.

� � 12 �8

x y

3 �7 1 5

b. 2x � y � 3z � �7x � 3y � 4z � 157x � 2y � z � �28

� � �7 15 �28

x y z

2 �1 3 1 3 �4 7 2 1

ExampleExample

ExercisesExercises


1. 2x � y � 8 2. 4x � 3y � 18 3. 7x � 2y � 155x � 3y � �12 x � 2y � 12 3x � y � �10

� � � � � �

4. 4x � 6y � 20 5. 5x � 2y � 18 6. 3x � y � 243x � y � 8� 0 x � �4y � 25 3y � 80 � 2x

� � � � � �

7. 2x � y � 7z � 12 8. 5x � y � 7z � 325x � y � 3z � 15 x � 3y � 2z � �18x � 2y � 6z � 25 2x � 4y � 3z � 12

� � � �

9. 4x � 3y � z � �100 10. x � 3y � 7z � 272x � y � 3z � �64 2x � y � 5z � 485x � 3y � 2z � 8 4x � 2y � 3z � 72

� � � �

11. 2x � 3y � 9z � �108 12. z � 45 � 3x � 2yx � 5z � 40 � 2y 2x � 3y � z � 603x � 5y � 89� 4z x � 4y � 2z � 120

� � � � 45 60120

xyz

3 �2 12 3 �11 �4 2

�108 40 89

xyz

2 3 �91 �2 53 5 �4

274872

xyz

1 �3 72 1 �54 �2 3

�100 �64 8

xyz

4 �3 �12 1 �35 3 �2

32�18 12

xyz

5 �1 71 3 �22 4 �3

121525

xyz

2 1 75 �1 31 2 �6

2480

xy

3 �12 3

1825

xy

5 21 4

20�8

xy

4 �63 1

15�10

xy

7 �23 1

1812

xy

4 �31 2

8�12

xy

2 15 �3


Solve Systems of Equations Use inverse matrices to solve systems of equationswritten as matrix equations.

Solving Matrix EquationsIf AX � B, then X � A�1B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.

Solve � .

In the matrix equation A � , X � , and B � .

Step 1 Find the inverse of the coefficient matrix.

A�1 � or .

Step 2 Multiply each side of the matrix equation by the inverse matrix.

� � � � Multiply each side by A�1.

� � Multiply matrices.

� Simplify.

The solution is (2, �2).

Solve each matrix equation or system of equations by using inverse matrices.

1. � � 2. � � 3. � �

(5, �3) no solution (13, �18)

4. � � 5. � � 6. � �

�� , � � (57, �31) �� , �7. 4x � 2y � 22 8. 2x � y � 2 9. 3x � 4y � 12

6x � 4y � �2 x � 2y � 46 5x � 8y � �8

(3, �5) (10, 18) (32, �21)

10. x � 3y � �5 11. 5x � 4y � 5 12. 3x � 2y � 52x � 7y � 8 9x � 8y � 0 x � 4y � 20

(�59, 18) � , � (�2, �5.5)45�76

10�19

15�7

9�7

3�2

1�4

3 �6

x y

1 2 3 �1

�15 6

x y

3 6 5 9

4 �8

x y

2 �3 2 5

3 �7

x y

3 2 5 4

16 12

x y

�4 �8 6 12

�2 18

x y

2 4 3 �1

2 �2

x y

16 �16

1�8

x y

1 0 0 1

6 4

4 �2 �6 5

1�8

x y

5 2 6 4

4 �2 �6 5

1�8

4 �2 �6 5

1�8

4 �2 �6 5

1�20 � 12

6 4

x y

5 2 6 4

64

x y

5 26 4


Using Matrices to Solve Systems of Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

4-84-8

ExampleExample

ExercisesExercises

Skills PracticeUsing Matrices to Solve Systems of Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

4-84-8


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1. x � y � 5 2. 3a � 8b � 162x � y � 1 4a � 3b � 3

� � � �

3. m � 3n � �3 4. 2c � 3d � 64m � 3n � 6 3c � 4d � 7

� � � �

5. r � s � 1 6. x � y � 52r � 3s � 12 3x � 2y � 10

� � � �

7. 6x � y � 2z � �4 8. a � b � c � 5�3x � 2y � z � 10 3a � 2b � c � 0x � y � z � 3 2a � 3b � 8

� � � �


9. � � (2, �3) 10. � � (3, �2)

11. � � (3, �2) 12. � � (3, 2)

13. � � �7, � 14. � � �1, �

15. p � 3q � 6 16. �x � 3y � 22p � 3q � �6 (0, �2) �4x � 5y � 1 (1, �1)

17. 2m � 2n � �8 18. �3a � b � �96m � 4n � �18 (�1, �3) 5a � 2b � 14 (4, 3)

5�3

15 2

m n

5 6 12 �6

1�3

25 12

c d

3 12 2 �6

15 23

m n

7 �3 5 4

�1 7

a b

5 8 3 1

6 �3

x y

4 3 1 3

�7 �1

w z

1 3 4 3

508

abc

1 �1 13 2 �12 3 0

�4 10 3

xyz

6 �1 2�3 2 �1 1 1 1

510

xy

1 13 2

112

rs

1 �12 3

67

cd

2 33 �4

�3 6

m n

1 34 3

16 3

ab

3 84 3

51

xy

1 12 �1



1. �3x � 2y � 9 2. 6x � 2y � �25x � 3y � �13 3x � 3y � 10

� � � �

3. 2a � b � 0 4. r � 5s � 103a � 2b � �2 2r � 3s � 7

� � � �

5. 3x � 2y � 5z � 3 6. 2m � n � 3p � �5x � y � 4z � 2 5m � 2n � 2p � 8�2x � 2y � 7z � �5 3m � 3n � 5p � 17

� � � �


7. � � (2, �4) 8. � � (5, 1)

9. � � (3, �5) 10. � � (1, 7)

11. � � ��4, � 12. � � �� , �13. 2x � 3y � 5 14. 8d � 9f � 13

3x � 2y � 1 (1, 1) �6d � 5f � �45 (5, �3)

15. 5m � 9n � 19 16. �4j � 9k � �82m � n � �20 (�7, 6) 6j � 12k � �5 � , � �

17. AIRLINE TICKETS Last Monday at 7:30 A.M., an airline flew 89 passengers on acommuter flight from Boston to New York. Some of the passengers paid $120 for theirtickets and the rest paid $230 for their tickets. The total cost of all of the tickets was$14,200. How many passengers bought $120 tickets? How many bought $230 tickets?57; 32

18. NUTRITION A single dose of a dietary supplement contains 0.2 gram of calcium and 0.2 gram of vitamin C. A single dose of a second dietary supplement contains 0.1 gram of calcium and 0.4 gram of vitamin C. If a person wants to take 0.6 gram of calcium and1.2 grams of vitamin C, how many doses of each supplement should she take?2 doses of the first supplement and 2 doses of the second supplement

2�3

1�2

1�3

1�4

�1 �1

y z

8 3 12 6

1�2

17 �26

r s

�4 2 7 4

16 34

c d

�5 3 6 4

12 �11

a b

�1 �3 3 4

�7 10

x y

�2 3 1 5

0 �2

g h

2 1 3 2

�5 8 17

m n p

2 1 �35 2 �23 �3 5

3 2�5

xyz

3 �2 5 1 1 �4�2 2 7

10 7

rs

1 52 �3

0�2

ab

2 13 2

�2 10

xy

6 �23 3

9�13

xy

�3 2 5 �3

Practice (Average)

Using Matrices to Solve Systems of Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

4-84-8

Reading to Learn MathematicsUsing Matrices to Solve Systems of Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

4-84-8


Less

on

4-8

Pre-Activity How are inverse matrices used in population ecology?


Write a 2 � 2 matrix that summarizes the information given in theintroduction about the food and territory requirements for the two species.

Reading the Lesson

1. a. Write a matrix equation for the following system of equations.3x � 5y � 10

� � 2x � 4y � �7

b. Explain how to use the matrix equation you wrote above to solve the system. Use asfew mathematical symbols in your explanation as you can. Do not actually solve thesystem.

Sample answer: Find the inverse of the 2 � 2 matrix of coefficients.Multiply this inverse by the 2 � 1 matrix of constants, with the 2 � 2matrix on the left. The product will be a 2 � 1 matrix. The number inthe first row will be the value of x, and the number in the second rowwill be the value of y.

2. Write a system of equations that corresponds to the following matrix equation.

� �3x � 2y � 4z � �22x � y � 65y � 6z � �4


3. Some students have trouble remembering how to set up a matrix equation to solve asystem of linear equations. What is an easy way to remember the order in which to writethe three matrices that make up the equation?

Sample answer: Just remember “CVC” for “coefficients, variables,constants.”The variable matrix is on the left side of the equals sign, justas the variables are in the system of linear equations. The constantmatrix is on the right side of the equals sign, just as the constants are inthe system of linear equations.

�2 6 �4

x y z

3 2 �4 2 �1 0 0 5 6

10�7

xy

3 52 �4

140 500120 400


Properties of MatricesComputing with matrices is different from computing with real numbers. Stated below are some properties of the real number system. Are these also true for matrices? In the problems on this page, you will investigate this question.

For all real numbers a and b, ab � 0 if and only if a � 0 or b � 0.

Multiplication is commutative. For all real numbers a and b, ab � ba.

Multiplication is associative. For all real numbers a, b, and c, a(bc) � (ab)c.

Use the matrices A, B, and C for the problems. Write whether each statement is true. Assume that a 2-by-2 matrix is the 0 matrix if and only if all of its elements are zero.

A � B � C �

1. AB � 0 2. AC � 0 3. BC � 0

4. AB � BA 5. AC � CA 6. BC � CB

7. A(BC) � (AB)C 8. B(CA) � (BC)A 9. B(AC) � (BA)C

10. Write a statement summarizing your findings about the properties of matrixmultiplication.Based on these examples, matrix multiplication is associative, but notcommutative. Two matrices may have a product of zero even if neither ofthe factors equals zero.

�8 �

3 6 1 2

1 �3 �1 3

3 1 1 3

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

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Chapter 4 Test, Form 1

NAME DATE PERIOD

SCORE


Ass

essm

ent

Write the letter for the correct answer in the blank at the right of each question.

1. How many elements are there in a 3 � 4 matrix?A. 7 B. 3 C. 12 D. 4 1.

2. Solve the matrix equation � � � � � for y.

A. 1 B. 6 C. 7 D. �1 2.

For Questions 3–10, use the matrices to find the following.

P � � � Q � � � R � � � S � � �T � � � U � � � V � � �

3. the first row of T � UA. [�4 1 12] B. [1 �8 �9] C. [16 �9 6] D. not possible 3.

4. the first row of U � VA. [10 �3 �8] B. [�10 2] C. [10 �7 2] D. not possible 4.

5. the first row of 4TA. [�2 8 �5] B. [12 �4 �20] C. [24 �16 36] D. not possible 5.

6. the first row of 2P � SA. [9 �1] B. [10 �4] C. [9 5] D. not possible 6.

7. the first row of TVA. [12 �4 �20] B. [�23 21] C. [53 �27] D. not possible 7.

8. the inverse of matrix RA. P B. Q C. S D. not possible 8.

9. the dimensions of PQA. 1 � 2 B. 2 � 2 C. 2 � 1 D. 4 � 4 9.

10. the pair of matrices that are inversesA. P and S B. Q and S C. P and Q D. Q and R 10.

11. Find the value of � �.A. 13 B. 7 C. 17 D. 3 11.

12. Which expression is true for all matrices X2�2, Y2�2 and scalars c?A. c(X � Y) � (Y � X)c B. XY � YXC. c(XY) � (YX)c D. c(XY) � (cX)(cY) 12.

5 13 2

0 4�2 6

5 �3

10 �5 �32 7 4

6 �4 93 �1 �5

1 �30 �

12�

0 �12�

1 �21 60 2

4 12 0

3 61 z

3 xy 7

44


Chapter 4 Test, Form 1 (continued)

13. Evaluate � � using expansion by minors.

A. 5 B. �7 C. 7 D. �3 13.

14. Evaluate � � using diagonals.

A. �2 B. 7 C. 11 D. �1 14.

15. Triangle RST with vertices R(2, 5), S(1, 4), and T(3, 1) is translated 3 units right. What are the coordinates of S�?A. (4, 4) B. (4, 7) C. (1, 7) D. (�2, 4) 15.

16. The vertices of XYZ are X(3, 1), Y(0, 4), and Z(0, 0). The triangle is being

reflected over the line y � x. Use the reflection matrix � � to find X�.

A. (3, 1) B. (1, 3) C. (�3, 1) D. (3, �1) 16.

17. Cramer’s Rule is used to solve the system of equations 2m � 3n � 11 and 3m � 5n � 6. Which determinant represents the numerator for m?

A. � � B. � � C. � � D. � � 17.

18. Cramer’s Rule is used to solve the system of equations 2x � 3y � 4z � 12,3x � y � 5z � 10 and x � 4y � z � 8. Which determinant represents the numerator for y?

A. � � B. � � C. � � D. � � 18.

19. Which matrix would not be used to write a matrix equation for the system of equations 3f � 2g � 7 and �2f � g � �5?

A. � � B. � � C. � � D. � � 19.

20. Which product would be used to solve the matrix equation � � � � � � � �by using inverse matrices?

A. � � � � � B. �14�� C. �

14�� D. 4� � � � � 20.

Bonus Find the value of � �. B:0 1 0a b cc a b

40

1 �60 4

40

4 60 1

40

1 �60 4

40

4 60 1

40

mn

4 60 1

7�5

gf

3 �2�2 1

fg

2 4 123 5 101 �1 8

2 �3 43 1 51 �4 �1

12 �3 410 1 58 �4 �1

2 12 43 10 51 8 �1

11 36 �5

2 113 6

2 33 �5

11 26 3

0 11 0

2 0 13 1 21 �2 5

1 3 20 �1 12 4 1

NAME DATE PERIOD

44

Chapter 4 Test, Form 2A

NAME DATE PERIOD

SCORE


Ass

essm

ent


1. Solve the matrix equation � � � � � for x.

A. 20 B. 25 C. 1 D. 4 1.


P � � � Q � � � R � � � S � � �T � � � U � � � V � � �

2. the dimensions of matrix TA. 6 � 1 B. 3 � 2 C. 2 � 3 D. 1 � 6 2.

3. the first row of T � VA. [�9 6 8] B. [�9 �2] C. [�7 7 4] D. not possible 3.

4. the first row of U � TA. [14 �7 �7] B. [0] C. [10 �6 �13] D. not possible 4.

5. the first row of �4UA. [�4] B. [�16 36 20] C. [�12 0 8] D. not possible 5.

6. the first row of 4Q � PA. [4 �2] B. [4 �6] C. [4 22] D. not possible 6.

7. the first row of UVA. [2 21] B. [22 �36 �24] C. [0 14] D. not possible 7.


9. the dimensions of VTA. 2 � 3 B. 3 � 2 C. 3 � 3 D. 2 � 2 9.

10. the pair of matrices that are inversesA. P and Q B. P and S C. Q and S D. P and R 10.

11. Find the value of � �.A. 8 B. �64 C. 6 D. �8 11.

12. GEOMETRY Find the area of a triangle whose vertices have coordinates (�4, 3), (2, 5), and (�7, �1).A. 9 units2 B. 18 units2 C. 14 units2 D. 7 units2 12.

12 �47 �3

2 4�1 0

3 �1

3 0 �24 �9 �5

�11 7 5�6 �3 8

��34� �

12�

�114� 0

1 �12�

0 ��14�

1 20 �4

0 142 21

10 � 2y5 � x

3xy

44


Chapter 4 Test, Form 2A (continued)


A. 58 B. �2 C. 12 D. 10 13.

14. For all matrices X3�3, Y2�3, and Z3�3 and scalars q, which statement is always true?A. X � 2Z � 2X � Z B. q(XZ) � (qX)ZC. q(YZ) � (ZY)q D. (XY)Z � Z(YX) 14.

15. MAP On a map, the coordinates of the corners of a town are A(0.5, 2),B(2, 3.5), C(5, 1.5), and D(3, 1). The map is dilated so that the perimeter of the town is five times its original perimeter. Find the coordinates of C�.A. (25, 7.5) B. (1, 0.3) C. (25, 1.5) D. (10, 6.5) 15.

16. Triangle RST with vertices R(�10, �8), S(1, 7), and T(5, �10) is rotated 90�counterclockwise about the origin. Find the coordinates of T�.A. (�5, 10) B. (�10, �5) C. (10, 5) D. (10, �5) 16.

17. Cramer’s Rule is used to solve the system of equations 3m � 5n � 12 and 4m � 7n � �5. Which determinant represents the numerator for n?

A. � � B. � � C. � � D. � � 17.

18. Cramer’s Rule is used to solve the system of equations 3x � y � 2z � 17,4x � 2y � 3z � 10, and 2x � 5y � 9z � �6. Which determinant represents the numerator for z?

A. � � B. � � C. � � D. � � 18.

19. Which matrix would not be used to write a matrix equation for the system of equations 5m � 2n � 13 and �m � n � �2?

A. � � B. � � C. � � D. � � 19.

20. Which product would be used to solve the matrix equation

� � � � � � � � by using inverse matrices?

A. �111�� B. �1

11�� C. � � � � � D. � � � � � 20.

Bonus Find the value of � �. B:d f d1 1 1f d f

67

3 �42 1

67

1 4�2 3

67

3 �42 1

67

1 4�2 3

67

ab

3 �42 1

5 �2�1 1

5 13�1 �2

13�2

mn

3 17 24 10 �32 �6 �9

17 3 �110 4 2

�6 2 5

3 �1 174 2 102 5 �6

3 �1 24 2 �32 5 �9

12 �5�5 7

3 124 �5

3 �54 7

12 3�5 4

1 4 �10 3 52 6 �2

NAME DATE PERIOD

44

Chapter 4 Test, Form 2B

NAME DATE PERIOD

SCORE


Ass

essm

ent


1. Solve the matrix equation � � � � � for x.

A. 1 B. �1 C. 7 D. �7 1.


P � � � Q � � � R � � � S � � �T � � � U � � � V � � �

2. the dimensions of matrix VA. 6 � 1 B. 1 � 6 C. 2 � 3 D. 3 � 2 2.

3. the first row of T � UA. [�5 1 6] B. [13 �11 �2] C. [3 �3 6] D. not possible 3.

4. the first row of V � TA. [�1 �7] B. [�7 3 2] C. [�1 5 �6] D. not possible 4.

5. the first row of �3TA. [�12 �24] B. [�12 15 �6] C. [�24 3 �9] D. not possible 5.

6. the first row of 5P � QA. [11 1] B. [9 1] C. [19 9] D. not possible 6.

7. the first row of TVA. [20 �16 9] B. [20 24] C. [4 4] D. not possible 7.


9. the dimensions of STA. 1 � 6 B. 2 � 2 C. 3 � 2 D. 2 � 3 9.

10. the pair of matrices that are inversesA. S and R B. Q and R C. P and Q D. Q and S 10.

11. Find the value of � �.A. 6 B. �6 C. �54 D. 54 11.

12. GEOMETRY Find the area of a triangle whose vertices have coordinates (2, �5), (6, 1), and (�3, �4).A. 34 units2 B. 21 units2 C. 17 units2 D. 42 units2 12.

15 4�6 �2

3 10 2

�4 5

�9 6 4�5 �2 3

4 �5 28 �1 3

�172� ��

19�

��13� �

19�

0 ��14�

1 �34�

4 412 21

3 1�4 0

2x3y

44


Chapter 4 Test, Form 2B (continued)


A. –38 B. 94 C. �42 D. 114 13.

14. For all matrices A2�2, B2�4, and C2�2, and scalars d, which statement is always true?A. A � dC � dA � C B. (CB)d � d(BC)C. d(A � C) � dA � dC D. (A � C)B � B(A � C) 14.

15. MAP On a town map, the coordinates of the corners of the zoo are L(1.2, 4),M(2, 0.8), N(4, 1.6), and P(6, 6). The map is dilated so that the perimeter of the zoo is four times its original size. Find the coordinates of M�.A. (8, 0.8) B. (0.5, 0.1) C. (8, 3.2) D. (6, 4.8) 15.

16. Triangle EFG with vertices E(�4, �5), F(1, 3), and G(4, �1) is rotated 270�counterclockwise about the origin. Find the coordinates of G�.A. (1, 4) B. (�4, 1) C. (�1, 4) D. (�1, �4) 16.

17. Cramer’s Rule is used to solve the system of equations 5f � 9g � 10 and 4f � 3g � �6. Which determinant represents the numerator for f?

A. � � B. � � C. � � D. � � 17.

18. Cramer’s Rule is used to solve the system of equations 4x � 5y � z � 11,3x � 2y � 2z � �5, and 2x � 6y � 3z � 8. Which determinant represents the numerator for z?

A. � � B. � � C. � � D. � � 18.

19. Which matrix would not be used to write a matrix equation for the system of equations 2c � 5d � �11 and �c � 2d � 10?

A. �19�� B. � � C. � � D. � � 19.

20. Which product would be used to solve the matrix equation

� � � � � � � � by using inverse matrices?

A. � � � � � B. �110�� C. �1

10�� D. � � � � � 20.

Bonus Find the value of � �. B:�a b �c

a �b ca 1 1

26

7 �31 1

26

7 �31 1

26

1 3�1 7

26

1 3�1 7

26

mn

7 �31 1

�1110

2 5�1 2

cd

2 �51 2

4 �5 13 �2 22 6 3

4 �5 113 �2 �52 6 8

4 11 �53 �5 �22 8 6

11 4 �5�5 3 �2

8 2 6

�9 103 �6

5 104 �6

5 �94 3

10 �9�6 3

2 �3 14 0 �25 �1 6

NAME DATE PERIOD

44

Chapter 4 Test, Form 2C


1. GOLF At a municipal golf course, the fees to play a round 1.of golf are as shown. Write a 3 � 2 matrix to organize this information.

2. Solve the matrix equation � � � � �. 2.

For Questions 3–6, perform the indicated matrix operations. If the matrix does not exist, write impossible.

3. � � � � � 3.

4. � � � � � 4.

5. 4[2 1 9 12] � 2[�5 8 7 0] 5.

6. 7� � � 5� � � 3� � 6.

7. Determine whether the matrix X3�5 � Y5�9 is defined. 7.If so, state the dimensions of the product.

8. Find � � � � �, if possible. 8.

9. Use A � � �, B � � �, and C � � � to find 9.

(AB)C and A(BC). Then state whether (AB)C � A(BC) is true for the given matrices.

10. Triangle ABC with vertices A(�4, �3), B(2, �5), and 10.C(�1, 4) is translated 3 units right and 2 units down. Find the coordinates of �A�B�C�.

0 �27 3

5 �3�6 1

3 10 �4

9 �8 12 �4 �6

4 �12 �5

104

�1

61

�2

30

�4

3 12 �117 2 8

1 �4 9 03 2 �5 6

10 �5 3 19 �4 0 �2

6 3 10 41 �7 �8 3

18�11

2x � 3y3x � 5y

NAME DATE PERIOD

SCORE 44

Ass

essm

ent

Resident Non-resident

Weekend

Weekday morning $20 $25

$40

Weekday afternoon $18 $22

$40


Chapter 4 Test, Form 2C (continued)

11. The vertices of parallelogram RSTU are R(�4, �2), S(0, �2), 11.T(6, 1), and U(2, 1). The parallelogram is reflected over the y-axis. Find the coordinates of parallelogram R�S�T�U�.

12. Find the value of � �. 12.

13. Evaluate � � using expansion by minors. 13.

14. GEOMETRY Find the area of a triangle whose vertices 14.are located at (�7, �1), (1, 6), and (5, �3).

For Questions 15 and 16, use Cramer’s Rule to solve each system of equations.

15. 3a � 2b � 6.5 15.2a � 1.5b � 10

16. 2x � 5y � 3z � 27 16.4x � 3y � 7z � �37x � 2y � 5z � 30

17. Determine whether P � � � and Q � � � are 17.

inverses.

18. Find the inverse of R � � �, if it exists. 18.

19. Solve the matrix equation � � � � � � � � by using 19.

inverse matrices.

20. TRANSPORTATION A transportation company rents cars, 20.trucks, and vans. The company has 366 vehicles in its fleet.It has twice as many trucks as vans, and 186 more cars than vans. Let c represent the number of cars, t represent the number of trucks, and v represent the number of vans.Write a matrix equation that describes this situation.

Bonus Find the value of � �. B:x �y 1

�x �y 11 0 1

54

xy

�2 4�1 3

4 22 4

��16� �

14�

�12� ��

14�

3 36 2

6 1 4�5 9 �3

2 �8 4

�7 124 �11

NAME DATE PERIOD

44

Chapter 4 Test, Form 2D


1. TICKETS At a local ballpark, ticket prices for home games 1.are as shown. Write a 3 � 2 matrix to organize this information.



3. � � � � � 3.

4. � � � � � 4.

5. 9[�1 3 7 2] � 6[4 5 �8 1] 5.

6. 4� � � 3� � � 6� � 6.

7. Determine whether the matrix P2�4 � Q4�7 is defined. 7.If so, state the dimensions of the product.

8. Find � � � � �, if possible. 8.

9. Use A � � �, B � � �, and scalar d � �2 to find 9.

d(AB) and (dA)B. Then state whether d(AB) � (dA)B is true for the given matrices.

10. Triangle ABC with vertices A(4, �7), B(2, 3), and C(�3, �1) 10.is translated 2 units left and 4 units up. Find the coordinates of �A�B�C�.

11 �72 1

4 9�5 2

3 �4 �98 7 10

2 1�3 �6

�137

92

�1

0�5

3

12 �6 98 4 5

�7 �3 9

4 �51 69 �2

�6 3 0 114 �7 12 5

9 �6 10 17 4 �8 0

�3412

3x � 7y4x � 5y

NAME DATE PERIOD

SCORE 44

Ass

essm

ent

Bleachers Box Seats

Double-Header

Weekday $9 $21

$27

Weekend $12 $26

$27


Chapter 4 Test, Form 2D (continued)

11. The vertices of parallelogram RSTU are R(�3, �5), S(4, �5), 11.T(6, 4), and U(�1, 4). The parallelogram is reflected over the x-axis. Find the coordinates of parallelogram R�S�T�U�.



14. GEOMETRY Find the area of a triangle whose vertices are 14.located at (�8, 4), (2, 6), and (3, �2).


15. 0.5x � 2y � 7 15.3x � 10y � 9

16. 4a � 3b � 5c � 2 16.2a � 4b � 7c � 20a � 3b � 8c � �13

17. Determine whether S � � � and T � � � are inverses. 17.

18. Find the inverse of A � � �, if it exists. 18.

19. Solve the matrix equation � � � � � � � � by using 19.

inverse matrices.

20. GARDENING A garden shop sells flowers, trees, and shrubs. 20.The shop sold 1085 items last month. Fifteen more shrubs were sold than trees, and 8 times as many flowers were sold as shrubs. Let f represent the number of flowers, t represent the number of trees, and s represent the number of shrubs.Write a matrix equation that describes this situation.

Bonus Find the value of � �. B:a b �c1 1 0a �b c

�72

xy

6 �8�1 3

4 �23 �1

�112� �

13�

�16� �

13�

4 42 �1

9 �5 81 2 14 �2 �2

10 �76 �5

NAME DATE PERIOD

44

Chapter 4 Test, Form 3


1. TICKETS A theater group is to put on four performances 1.of its latest production. Each ticket for a Friday or Saturday evening show will cost $10. Each ticket for a Saturday or Sunday matinee will cost $7. Write a 3 � 2 matrix to organize this information.



3. � � � � � 3.

4. �16��

14�� 4.

5. 3� � � 5� � 5.

6. � � � � � 6.

7. Use A � � �, B � � �, and C � � � to find 7.

C(BA) and (AB)C. Then state whether C(BA) � (AB)C is true for the given matrices.

8. Quadrilateral WXYZ with vertices W(�4, 2), X(�1, 4), 8.Y(6, �1), and Z(�2, �3) is translated so that W� is at (5, �2).Find the coordinates of X�, Y�, and Z�.

9. A triangle is rotated 270� counterclockwise about the origin. 9.The coordinates of the new vertices are R�(�4, 1), S�(1, �8),and T�(6, 4). What were the coordinates of the triangle in its original position?


11. Solve for x if � � � 18. 11.�4 2x3 x

6.0 �3.1�5.9 �4.8

0 �42 �1

�1 ��32��4 �3

2 2

4 �2 ��12�

6 1 0

�1 �12�

0 4

�1 3 �14�

�25� �1 0

��12� �1 0

3 �15� 2

8 21 �1

12 0�6 3

0.95 �7.243.59 3.412.68 �5.01

2.37 4.121.69 3.97

�5.18 6.25

3 x � 11 4

x2 � 1 4 � y2x � y y � 1

NAME DATE PERIOD

SCORE 44

Ass

essm

ent


Chapter 4 Test, Form 3 (continued)


13. GEOMETRY Find the area of a triangle whose vertices are 13.

located at ��4, �12��, ��

52�, �1�, and (6, �2). Evaluate the

determinant using diagonals.


14. �12�x � �

14�y � 1

3x � �12�y � �4

16. Determine whether M � � � and N � � � are 16.

inverses.

17. Find the inverse of A � � �, if it exists. 17.

For Questions 18 and 19, solve each system of equations by using inverse matrices.

18. 9a � 4b � �5 18.6a � 2b � �3

19. 5x � 3y � �4.5 19.2x � 1.2y � �1.8

20. BUDGET A foundation spent all of its operating budget 20.on administrative expenses, renewal grants, and low-interest loans. Administrative expenses and renewal grants accounted for 78.4% of the total budget. Renewable grants accounted for 5.1% more of the total budget than low-interest loans. Let a,r, and l each represent the percent of the budget accounted for by administrative expenses, renewal grants, and low-interest loans, respectively. Write a matrix equation that describes this situation.

Bonus Find the value of � �. B:a � 1 a 1b � 1 b �1c � 1 c 1

�15� �

23�

��25� ��

13�

�25� 1

�15� 1

5 �5�1 2

15. 3m � 4n � 6p � 152m � 3n � 5p � �115m � 6n � p � 9

7 5 4�3 �9 5

2 0 �3

NAME DATE PERIOD

44

14.

15.

Chapter 4 Open-Ended Assessment


Demonstrate your knowledge by giving a clear, concise solutionto each problem. Be sure to include all relevant drawings andjustify your answers. You may show your solutions in more thanone way or investigate beyond the requirements of the problem.

1. State any conditions or restrictions on m, n, j, and k when performing the indicated operations for matrices Am�n and Bj�k.a. adding A and Bb. multiplying A and Bc. finding the determinant of Ad. multiplying B by a scalar ce. finding the inverse of A

2. Choose three points in the coordinate plane so that no two points lie in the same quadrant. Consider these points the vertices of �XYZ.a. Write the vertex matrix V for �XYZ, and state the dimensions of V.

For parts b through f, use T � � �, R � � �, C � � �,and I � � � to find each sum or product. Then state the type of

transformation represented by the new matrix, describing the relationship between the image and premimage of �XYZ.

b. T � V c. 3V d. RV e. CV f. IV

3. Suppose you are asked to find the value of a if � � � 0.

a. If you must evaluate the determinant above using expansion by minors,which row would you use for the expansion? Explain your reasoning.

b. If you were able to choose which method, either using expansion by minors or using diagonals, to evaluate this determinant, which method would you choose? Why?

c. Find the value of a using the method of your choice.

4. Christopher purchased three CDs and two videos, and spent a total of $85. Edward purchased two CDs and one video, spending $50.a. Write a system of equations that describes this situation. Explain

what the variables in your system represent.b. Use Cramer’s Rule to solve your system. Interpret the meaning of

your solution.c. Write a matrix equation for your system of equations. Then solve

the matrix equation by using inverse matrices.d. Which of the two methods do you prefer. Why?

2 �7 57 a �21 0 �1

1 00 1

0 �11 0

�1 00 1

3 3 3�4 �4 �4

NAME DATE PERIOD

SCORE 44

Ass

essm

ent


Chapter 4 Vocabulary Test/Review

Underline or circle the correct word or phrase to complete each sentence.

1. (Scalar multiplication, Translation, Cramer’s Rule) is a methodfor solving systems of equations that uses determinants.

2. A (row matrix, square matrix, column matrix) is a matrix withthe same number of rows as columns.

3. Any function that maps points of a preimage onto its image iscalled a(n) (transformation, isometry, dilation).

4. A rectangular array of variables or constants is called a(n)(scalar, matrix, element).

5. A (reflection, rotation, translation) is a transformation in whichevery point of a figure is mapped to a corresponding image acrossa line of symmetry.

6. (Cramer’s Rule, Scalar multiplication, Expansion by minors) is amethod for evaluating a third-order determinant.

7. If the row and column containing a specific element of a 3 � 3matrix are deleted, the resulting 2 � 2 matrix is called the(inverse, zero matrix, minor) of that element.

8. A transformation in which the preimage and the image arecongruent figures is called a(n) (isometry, scalar, dilation).

9. For triangle ABC, the coordinates of points A, B, and C can bewritten in a (row matrix, vertex matrix, column matrix).

10. A transformation that enlarges or reduces a figure is called a(translation, dilation, rotation).

In your own words—Define each term.

11. identity matrix

12. column matrix

column matrixCramer’s Ruledeterminantdilationdimensionelementequal matrices

expansion by minorsidentity matriximageinverseisometrymatrixmatrix equation

minorpreimagereflectionrotationrow matrixscalarscalar multiplication

second-order determinantsquare matrixthird-order determinanttransformationtranslationvertex matrixzero matrix

NAME DATE PERIOD

SCORE 44

Ass

essm

ent

Chapter 4 Quiz (Lessons 4–1 and 4–2)

44


1. GRADES On the first two algebra exams of the year, 1.Trista’s grades were 80 and 95, Javier’s grades were 85 and 90, and Yolanda’s grades were 75 and 90. Write a 3 � 2 matrix to organize this information.


Use matrices A, B, and C to find the following.

A � � � B � � � C � � �

3. the dimensions of matrix A

4. B � C 5. 3C � 2B

1 7 �5�10 0 2

3 5 �812 �11 9

6 4 �8 51 �3 9 72 0 �2 �4

�29

x � 5y2x � 3y

NAME DATE PERIOD

SCORE


1. Determine whether the matrix product A5�2 � B2�4 is defined. 1.If so, state the dimensions of the product.

2. If possible, find the product � � � � �. 2.

3. Use A � � � and B � � �, and scalar c � 3 to find 3.

c(AB) and (BA)c. Then state whether the equation c(AB) � (BA)c is true for the given matrices.

4. Standardized Test Practice 4.Rectangle A�B�C�D� is the result of a translation of rectangle ABCD. A table of the translations is shown. Find the coordinates of A and D�.

5. The vertices of �XYZ are X(2, 3), Y(�3, 2), and Z(4, �5). The 5.triangle is being reflected over the line y � x. Find the coordinates of �X�Y�Z�.

�5 24 3

1 �24 3

2 �13 �31 4

1 5 �46 0 8

NAME DATE PERIOD

SCORE 44

3.

4.

5.

RectangleABCD

RectangleA�B�C�D�

A A� (1, 1)

B (1, �4) B� (4, 1)

C (1, 1) C� (4, 6)

D (�2, 1) D�




3. GEOMETRY Find the area of a triangle whose vertices 3.are located at (�2, 5), (�4, �3), and (3, 1). Evaluate the determinant using diagonals.


4. x � y � 6 4.3x � 2y � 11

5. 4x � 2y � z � �1 5.3x � 4y � 5z � �13x � 4y � 3z � 7

2 �1 34 0 1

�2 3 5

�3 42 �5


1. Determine whether A � � � and B � � � are 1.

inverses.

2. Find the inverse of M � � �, if it exists. 2.

For Questions 3 and 4, write a matrix equation for each system of equations.

3. a � 3b � 15 3.a � 4b � �13

4. 2x � 3y � 4z � �20 4.3x � z � 2x � 4y � z � �6

5. Solve the system of equations 3x � 2y � 22 and x � 2y � �6 5.by using inverse matrices.

�8 24 �1

�12� �

14�

0 ��12�

2 10 �2

NAME DATE PERIOD

SCORE


44

NAME DATE PERIOD

SCORE

44

Chapter 4 Mid-Chapter Test (Lessons 4–1 through 4–4)



A. 16 � 8 B. 2 � 2 � 3 C. 4 � 3 D. 3 � 4 1.

2. Solve the matrix equation � � � � � for y.

A. �1 B. 7 C. �3 D. 3 2.

3. For all matrices X3�5, Y2�3, Z3�4, and scalars c, which statement is always true?A. c(YZ) � (YZ)c B. YX � XYC. Y � Z � Z � Y D. c(ZX) � c(XY) 3.

4. Triangle RST with vertices R(�3, 4), S(0, �5), and T(6, 7) is translated 3 units left and 5 units down. Find the coordinates of R�.A. (�6, 1) B. (�6, �1) C. (0, �1) D. (0, 9) 4.

5. Parallelogram ABCD with A(�2, �2), B(4, �2), C(5, 3), and D(�1, 3) is rotated 270� counterclockwise about the origin. Find the coordinates of B�.A. (2, 4) B. (�4, 2) C. (2, �4) D. (�2, �4) 5.


6. � � � � � 6.

7. � � � � � 7.

8. �4� � 8.

9. 3[1 0 �9 6] � 4[5 12 0 �3] 9.

10. � � � � � 10.

11. � � � � � 11.4 6�2 7

�1 2 10 5 3

1 �23 �1

�4 01 5

3 �5 129 11 �7

�2 4 6

9 �7 �3 06 �5 1 19

0 6 13 �11�7 �4 2 4

4 �6 13 5�5 11 0 �4

3 12 0 79 �15 �8 6

Part II

117

3x � 2y4x � 5y

Part I

NAME DATE PERIOD

SCORE 44

Ass

essm

ent

1. State the dimensions of matrix F if F � � �.0 1 02 �4 24 �8 48 �16 8


Chapter 4 Cumulative Review (Chapters 1–4)

1. The formula S � �n(n

2� 1)� can be used to find the sum of the 1.

first n natural numbers. Find the sum of the first 100 natural numbers. (Lesson 1-1)

2. Evaluate 8 � � 3a � b � if a � 4.3 and b � �15. 2.(Lesson 1-4)

3. Find the slope of the line that passes through (4, –3) and 3.(–1, –7). (Lesson 2-3)

4. Identify the domain and range of the function g(x) � � x � �4. 4.(Lesson 2-6)

5. Graph y � 2x � � 1. (Lesson 2-7) 5.

6. The vertex of an angle is the point where the lines whose 6.equations are y � 2x and y � x � 1 meet. Find the coordinates of the vertex. (Lesson 3-1)

7. Determine whether the ordered pair (–2, 6) is a solution of 7.the system y 2x � 7 and y 5 � 3x. (Lesson 3-3)

8. Write a system of three equations to represent the given 8.information. The sum of three numbers is 37. The second number is 4 more than the first number, and the sum of the second number and the third number is 29. (Lesson 3-5)

9. Solve � � � � �. (Lesson 4-1) 9.

For Questions 10 and 11, perform the indicated matrix 10.operations. If the matrix does not exist, write impossible.

10. � � � � � 11. �4� � 11.

(Lesson 4-2) (Lesson 4-2)

12. Evaluate � � using diagonals. (Lesson 4-5) 12.

13. Find the inverse of M � � �, if it exists. (Lesson 4-7) 13.3 2�1 4

�4 2 �11 �1 2

�3 0 5

3 0 11�9 2 6

4 �3 �5

1 9 �5�7 6 4

3 1�2 17

�814

2x � yx � 4y

y

xO

NAME DATE PERIOD

44

Standardized Test Practice (Chapters 1–4)


1. If 4x2 � 13, then what is the value of (2x � 3)(2x � 3)?A. 4 B. 22 � 6�3� C. �13� � 9 D. –22 1.

2. If the measure of the edge of a cube is 3, the measure of the surface area of the cube is _____.E. 36 F. 54 G. 27 H. 9 2.

3. If the average of a and b equals the average of a, b, and c, then express c in terms of a and b.

A. a � b B. 2(a � b) C. �a �

2b

� D. �a �

3b

� 3.

4. There are 100 items in a garage sale. 30% of the items are sold during the first hour of the sale. If 10 items are sold during the second hour, what percent of the items have not been sold?E. 10% F. 60% G. 70% H. 90% 4.

5. If it takes 6 hours for 2 people to clean a house, how many hours will it take 4 people, working at the same rate, to clean another house that is the same size?

A. �23� B. 1�

12� C. 2 D. 3 5.

6. Which statement must be true?E. x y F. x � yG. x � y H. 2y � 180 � x 6.

7. Three less than three times a number is �56�. What is one more than

twice the number?

A. 2�23� B. 3�

49� C. 3�

59� D. 2�

12� 7.

8. r�s is defined as 2r2 � s2 � 4rs. What is the value of �2��1?E. –17 F. –1 G. 1 H. 17 8.

9. Which is closest to the value of �36.82.

�9

1.5�?

A. 19 B. 100 C. 150 D. 200 9.

10. In the figure, A�B� D�C�, DX � 8, and AD � 17.What is the length of BC?E. 17 F. 8�2�G. 15�2� H. 15 10. HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

NAME DATE PERIOD

44

Ass

essm

ent

Part 2: Grid In

Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.

2y˚(180 � x)˚

A B

D X CY

45˚


Standardized Test Practice (continued)

11. What is the value of r3 if �1r� � �0.01�? 11. 12.

12. If 5y�4 � 25 and 32x�1 � 27, what is the

value of �xy�?

13. For what integer value of m is 2m � 14 42

and 5 � �m4� � 3 � 7?

14. If D�C� is parallel to the 13. 14.x-axis and BC � 5 in the figure, what is the area of the shaded region?

Column A Column B

15. r 1 15.

16. a : b � c : d 16.

17. a 3, b � �4 17.

18. A � {7, 15, 2, 8, 14, 2} 18.

The mean of set A.The median of set A.

DCBA

�ba

� � 1�a �

ab

�

DCBA

bdac

DCBA

10r � 110r 2 � r

DCBA

Part 3: Quantitative Comparison

Instructions: Compare the quantities in columns A and B. Shade in if the quantity in column A is greater; if the quantity in column B is greater; if the quantities are equal; or if the relationship cannot be determined from the information given.

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

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87654321

0 0 0

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.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

NAME DATE PERIOD

44

Part 2: Grid In

Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.

y

xOC(10, 1)

A(�2, 5) B

D

A

D

C

B

Unit 1 Test(Chapters 1–4)


1. Find the value of 7 � 2(5 � 2 � 32). 1.

2. Name the sets of numbers to which 457 belongs. 2.

3. The sum of a number and 17 more than twice the same 3.number is 101. Find the number.

4. Evaluate 5 � � a � 5b � if a � �12 and b � 2. 4.

5. Define a variable and write an inequality. Then solve. 5.A local summer baseball team plays 20 games each season.So far, they have won 9 games and lost 2. How many more games must they win this season to win at least 75% of all their games?

6. Solve 3 � 2(1 � x) 6 or 2x � 14 8. Graph the solution set 6.on a number line.

7. Solve 3 � � 2y � 1 � � 1. Graph the solution set on a number 7.line.

8. If f(x) � �5x2

x� 4�, find f(4). 8.

9. Write an equation in slope-intercept form for the line that 9.passes through (3, 5) and (–2, 1).

10. Write an equation for the line that passes through (0, 7) and 10.

is perpendicular to the line whose equation is y � �12

�x � 1.

For Questions 11 and 12, use the set of data in the table.

The table shows the relationship between the price of a comic book and the number of copies sold. 11.

11. Draw a scatter plot for the data.

12. Use two ordered pairs to write a prediction equation. Then 12.use your prediction equation to predict the number of comic books sold when the price is $4.50.

13. Evaluate h��23

�� if h(x) � x � 2 �. 13.

14. Describe the system 6x � 2y � 10 and 9x � 3y � 8 as 14.consistent and independent, consistent and dependent, or inconsistent.

n

p

68

0

1012141618

1 2 3 4Price (dollars)

Num

ber

Sol

d

�4�5 �3 �2 �1 0 2 3 4 5 61

0 1�3 �2 �1�4

NAME DATE PERIOD

SCORE

Ass

essm

ent

Price p (in dollars) 2.00 2.50 2.75 3.00 3.50

Number sold n 16 13 12 10 7


Unit 1 Test (continued)(Chapters 1–4)

15. Solve the system of equations at 4x � 2y � 2 15.the right by using substitution. y � �4x � 7

16. Solve the system of equations at x � 3y � 5 16.the right by using elimination. 3x � y � 5

For Questions 17–19, use the following information.A furniture company displays bedroom sets which require 21 square meters of space and living room sets which require 42 square meters of space. The company, which has 546 square meters of available space, wants to display at least 6 bedroom sets and at least 5 living room sets. 17.

17. Let b represent the number of bedroom sets and � represent the number of living room sets. Write a system of inequalities to represent the number of furniture sets that can be displayed.

18. Draw the graph showing the feasible region. Label the 18.coordinates of the vertices of the feasible region.

19. If a bedroom set sells for $10,000 and a living room set sells 19.for $18,000, determine the number of bedroom sets and living room sets that must be sold to maximize the amount collected.

For Questions 20 and 21, use the matrices below.

A � � � B � � � C � � �20. Find A � B. 20.

21. Find BC, if possible. 21.

22. Triangle DEF with vertices D(2, 5), E(1, –6), and 22.F(–5, 3) is translated 3 units right and 2 units down. Find the coordinates of D�E�F�.


24. Use Cramer’s Rule to solve the system of equations 24.3x � 5y � 21 and 4x � 2y � 2.

25. Solve the matrix equation � � � � � � � � 25.

using inverse matrices.

32�5

mn

4 �51 2

12 5 �2�3 0 1�5 4 2

�12

�2

10 6 �7�4 3 0

17 2 311 4 �9

y

xO

4

8

12

4 8 12 16

NAME DATE PERIOD

Standardized Test PracticeStudent Record Sheet (Use with pages 216–217 of the Student Edition.)

© Glencoe/McGraw-Hill A1 Glencoe Algebra 2

NAME DATE PERIOD

44

An

swer

s

Select the best answer from the choices given and fill in the corresponding oval.

1 4 7 9

2 5 8 10

3 6

Solve the problem and write your answer in the blank.

For Questions 13–17, also enter your answer by writing each number or symbol ina box. Then fill in the corresponding oval for that number or symbol.

11 13 15 17

12

14 16

Select the best answer from the choices given and fill in the corresponding oval.

18 20 22

19 21 DCBADCBA

DCBADCBADCBA

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DCBADCBA

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DCBADCBADCBADCBA

Part 2 Short Response/Grid InPart 2 Short Response/Grid In

Part 1 Multiple ChoicePart 1 Multiple Choice

Part 3 Quantitative ComparisonPart 3 Quantitative Comparison


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4-1



Readin

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Math

em

ati

csIn

tro

du

ctio

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atri

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4-1

4-1

©G

lenc

oe/M

cGra

w-H

ill17

3G

lenc

oe A

lgeb

ra 2

Lesson 4-1

Pre-

Act

ivit

yH

ow a

re m

atri

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use

d t

o m

ake

dec

isio

ns?

Rea

d th

e in

trod

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Les

son

4-1

at

the

top

of p

age

154

in y

our

text

book

.

Wh

at i

s th

e ba

se p

rice

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a M

id-S

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SU

V?

$27,

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din

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on

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ive

the

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ensi

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ach

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a.2

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b.[

14

0�

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1 �

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2.Id

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s m

any

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owin

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scri

ptio

ns

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ly:r

ow m

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n m

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uar

e m

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54

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zero

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lum

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uar

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3.W

rite

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ual

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olve

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ay t

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lenc

oe/M

cGra

w-H

ill17

4G

lenc

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lgeb

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Tess

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n a

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his

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3-1

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3-3-

3-3-

64-

6-12

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126.

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7.4-

8-8

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3-3-

4-4

no

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sem

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rich

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4-1

4-1


An

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uid

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4-2

4-2

©G

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5G

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4-2

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4-2

4-2

©G

lenc

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cGra

w-H

ill17

7G

lenc

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lgeb

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8G

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Per

form

th

e in

dic

ated

mat

rix

oper

atio

ns.

If t

he

mat

rix

doe

s n

ot e

xist

,wri

teim

pos

sibl

e.

1.�

2.

�

3.�

3�

44.

7�

2

5.�

2�

4�

6.

�

Use

A�

,B�

,an

d C

�

to f

ind

th

e fo

llow

ing.

7.A

�B

8.A

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9.�

3B10

.4B

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11.�

2B�

3C12

.A�

0.5C

ECO

NO

MIC

SF

or E

xerc

ises

13

an

d 1

4,u

se t

he

tab

le t

hat

sh

ows

loan

s b

y an

eco

nom

icd

evel

opm

ent

boa

rd t

o w

omen

an

d m

en s

tart

ing

new

bu

sin

esse

s.

13.W

rite

tw

o m

atri

ces

that

re

pres

ent

the

nu

mbe

r of

new

busi

nes

ses

and

loan

am

oun

ts,

one

for

wom

en a

nd

one

for

men

.

,

14.F

ind

the

sum

of

the

nu

mbe

rs o

f n

ew b

usi

nes

ses

and

loan

am

oun

ts

for

both

men

an

d w

omen

ove

r th

e th

ree-

year

per

iod

expr

esse

d as

a

mat

rix.

15. P

ET N

UTR

ITIO

NU

se t

he

tabl

e th

at g

ives

nu

trit

ion

al

info

rmat

ion

for

tw

o ty

pes

of d

og f

ood.

Fin

d th

edi

ffer

ence

in

th

e pe

rcen

t of

pro

tein

,fat

,an

d fi

ber

betw

een

Mix

B a

nd

Mix

A e

xpre

ssed

as

a m

atri

x.[2

�4

3]

% P

rote

in%

Fat

% F

iber

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125

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B24

88

63

1,43

1,00

07

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574,

000

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0

36

864,

000

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672,

000

28

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000

27

567,

000

41

902,

000

35

777,

000

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men

Men

Bu

sin

esse

sL

oan

Bu

sin

esse

sL

oan

A

mo

un

t ($

)A

mo

un

t ($

)

1999

27$5

67,0

0036

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,000

2000

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e (

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e)

Op

erat

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s w

ith

Mat

rice

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

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ER

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____

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4-2

4-2

16

�15

62

14

4575


An

swer

s


Readin

g t

o L

earn

Math

em

ati

csO

per

atio

ns

wit

h M

atri

ces

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

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ER

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____

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4-2

4-2

©G

lenc

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cGra

w-H

ill17

9G

lenc

oe A

lgeb

ra 2

Lesson 4-2

Pre-

Act

ivit

yH

ow c

an m

atri

ces

be

use

d t

o ca

lcu

late

dai

ly d

ieta

ry n

eed

s?

Rea

d th

e in

trod

uct

ion

to

Les

son

4-2

at

the

top

of p

age

160

in y

our

text

book

.

•W

rite

a s

um

th

at r

epre

sen

ts t

he

tota

l n

um

ber

of C

alor

ies

in t

he

pati

ent’s

diet

for

Day

2.(

Do

not

act

ual

ly c

alcu

late

th

e su

m.)

482

�62

2 �

987

•W

rite

th

e su

m t

hat

rep

rese

nts

th

e to

tal

fat

con

ten

t in

th

e pa

tien

t’s d

iet

for

Day

3.(

Do

not

act

ual

ly c

alcu

late

th

e su

m.)

11 �

12 �

38

Rea

din

g t

he

Less

on

1.F

or e

ach

pai

r of

mat

rice

s,gi

ve t

he

dim

ensi

ons

of t

he

indi

cate

d su

m,d

iffe

ren

ce,o

r sc

alar

prod

uct.

If t

he in

dica

ted

sum

,dif

fere

nce,

or s

cala

r pr

oduc

t do

es n

ot e

xist

,wri

te i

mpo

ssib

le.

A�

B�

C�

D�

A�

D:

C�

D:

5B:

�4C

:2D

�3A

:

2.S

upp

ose

that

M,N

,an

d P

are

non

zero

2 �

4 m

atri

ces

and

kis

a n

egat

ive

real

nu

mbe

r.In

dica

te w

het

her

eac

h o

f th

e fo

llow

ing

stat

emen

ts i

s tr

ue

or f

alse

.

a.M

�(N

�P

) �

M�

(P�

N) tr

ue

b.

M�

N�

N�

Mfa

lse

c.M

�(N

�P

) �

(M�

N)

�P

fals

ed

.k

(M�

N)

�kM

�kN

tru

e

Hel

pin

g Y

ou

Rem

emb

er

3.T

he

mat

hem

atic

al t

erm

sca

lar

may

be

un

fam

ilia

r,bu

t it

s m

ean

ing

is r

elat

ed t

o th

e w

ord

scal

eas

use

d in

a s

cale

of

mil

eson

a m

ap.H

ow c

an t

his

usa

ge o

f th

e w

ord

scal

eh

elp

you

rem

embe

r th

e m

ean

ing

of s

cala

r?

Sam

ple

an

swer

:A

sca

le o

f m

iles

tells

yo

u h

ow

to

mu

ltip

ly t

he

dis

tan

ces

you

mea

sure

on

a m

ap b

y a

cert

ain

nu

mb

er t

o g

et t

he

actu

al d

ista

nce

bet

wee

n t

wo

loca

tio

ns.

Th

is m

ult

iplie

r is

oft

en c

alle

d a

sca

lefa

cto

r.A

scal

arre

pre

sen

ts t

he

sam

e id

ea:

It is

a r

eal n

um

ber

by

wh

ich

a m

atri

xca

n b

e m

ult

iplie

d t

o c

han

ge

all t

he

elem

ents

of

the

mat

rix

by a

un

iform

scal

efa

cto

r.

2 �

33

�2

2 �

2im

po

ssib

le2

�3

�3

60

�8

40

5

10

�3

6

412

�4

0

0�

5

35

6 �

28

1

©G

lenc

oe/M

cGra

w-H

ill18

0G

lenc

oe A

lgeb

ra 2

Su

nd

aram

’s S

ieve

Th

e pr

oper

ties

an

d pa

tter

ns

of p

rim

e n

um

bers

hav

e fa

scin

ated

man

y m

ath

emat

icia

ns.

In 1

934,

a yo

un

g E

ast

Indi

an s

tude

nt

nam

ed S

un

dara

m c

onst

ruct

ed t

he

foll

owin

g m

atri

x.

47

1013

1619

2225

..

.7

1217

2227

3237

42.

..

1017

2431

3845

5259

..

.13

2231

4049

5867

76.

..

1627

3849

6071

8293

..

..

..

..

..

..

..

A s

urp

risi

ng

prop

erty

of

this

mat

rix

is t

hat

it

can

be

use

d to

det

erm

ine

wh

eth

er o

r n

ot s

ome

nu

mbe

rs a

re p

rim

e.

Com

ple

te t

hes

e p

rob

lem

s to

dis

cove

r th

is p

rop

erty

.

1.T

he

firs

t ro

w a

nd

the

firs

t co

lum

n a

re c

reat

ed b

y u

sin

g an

ari

thm

etic

pa

tter

n.W

hat

is

the

com

mon

dif

fere

nce

use

d in

th

e pa

tter

n?

3

2.F

ind

the

nex

t fo

ur

nu

mbe

rs i

n t

he

firs

t ro

w.

28,3

1,34

,37

3.W

hat

are

th

e co

mm

on d

iffe

ren

ces

use

d to

cre

ate

the

patt

ern

s in

row

s 2,

3,4,

and

5?5,

7,9,

11

4.W

rite

th

e n

ext

two

row

s of

th

e m

atri

x.In

clu

de e

igh

t n

um

bers

in

eac

h r

ow.

row

6:

19,3

2,45

,58,

71,8

4,97

,110

;ro

w 7

:22

,37,

52,6

7,82

,97,

112,

127

5.C

hoo

se a

ny

five

nu

mbe

rs f

rom

th

e m

atri

x.F

or e

ach

nu

mbe

r n

,th

at y

ou

chos

e fr

om t

he

mat

rix,

fin

d 2n

�1.

An

swer

s w

ill v

ary.

6.W

rite

th

e fa

ctor

izat

ion

of

each

val

ue

of 2

n�

1 th

at y

ou f

oun

d in

pro

blem

5.

An

swer

s w

ill v

ary,

but

all n

um

ber

s ar

e co

mp

osi

te.

7.U

se y

our

resu

lts

from

pro

blem

s 5

and

6 to

com

plet

e th

is s

tate

men

t:If

n

occu

rs i

n t

he

mat

rix,

then

2n

�1

(is/

is n

ot)

a pr

ime

nu

mbe

r.

8.C

hoo

se a

ny

five

nu

mbe

rs t

hat

are

not

in

th

e m

atri

x.F

ind

2n�

1 fo

r ea

ch

of t

hes

e n

um

bers

.Sh

ow t

hat

eac

h r

esu

lt i

s a

prim

e n

um

ber.

An

swer

s w

ill v

ary,

but

all n

um

ber

s ar

e p

rim

e.

9.C

ompl

ete

this

sta

tem

ent:

If n

does

not

occ

ur

in t

he

mat

rix,

then

2n

�1

is

.a

pri

me

nu

mb

er

is n

ot

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-2

4-2



Stu

dy G

uid

e a

nd I

nte

rven

tion

Mu

ltip

lyin

g M

atri

ces

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-3

4-3

©G

lenc

oe/M

cGra

w-H

ill18

1G

lenc

oe A

lgeb

ra 2

Lesson 4-3

Mu

ltip

ly M

atri

ces

You

can

mu

ltip

ly t

wo

mat

rice

s if

an

d on

ly i

f th

e n

um

ber

of c

olu

mn

sin

th

e fi

rst

mat

rix

is e

qual

to

the

nu

mbe

r of

row

s in

th

e se

con

d m

atri

x.

Mu

ltip

licat

ion

of

Mat

rice

s�

�

Fin

d A

Bif

A�

and

B�

.

AB

�

� S

ubst

itutio

n

�M

ultip

ly c

olum

ns b

y ro

ws.

�S

impl

ify.

Fin

d e

ach

pro

du

ct,i

f p

ossi

ble

.

1.�

2.�

3.�

4.�

5.�

6.�

7.�

[04

�3]

8.�

9.�

no

t p

oss

ible

10

�16

1

8�

6

59

11

�21

1

3�

15

2

�2

3

1 �

24

2

0�

3

14

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�1

31

1

�3

�2

07

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5�

4

610

�

43

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7

24

�15

1

4�

17

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4

84

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17

26

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4

�1

�2

55

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2�

31

2 2

1

3�

2

04

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1

40

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11

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1

5�

2

7

�7

14

14

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2

34

12

3�

69

3�

1 2

43

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24

3

2 �

14

�1

0

37

30

03

4

1 �

23

�23

17

12

�10

�2

19

�

4(5)

�3(

�1)

�4(

�2)

�3(

3)

2(5

) �

(�2)

(�1)

2(�

2) �

(�2)

(3)

1(

5) �

7(�

1)1(

�2)

�7(

3)

5

�2

�1

3�

43

2

�2

1

7

5

�2

�1

3�

43

2�

2

1

7a

1x1

�b 1

x 2a 1

y 1�

b 1y 2

a

2x1

�b 2

x 2a 2

y 1�

b 2y 2

x

1y 1

x

2y 2

a

1b 1

a

2b 2

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill18

2G

lenc

oe A

lgeb

ra 2

Mu

ltip

licat

ive

Pro

per

ties

Th

e C

omm

uta

tive

Pro

pert

y of

Mu

ltip

lica

tion

doe

s n

oth

old

for

mat

rice

s.

Pro

per

ties

of

Mat

rix

Mu

ltip

licat

ion

For

any

mat

rices

A,

B,

and

Cfo

r w

hich

the

mat

rix p

rodu

ct is

defin

ed,

and

any

scal

ar c

, th

e fo

llow

ing

prop

ertie

s ar

e tr

ue.

Ass

oci

ativ

e P

rop

erty

of

Mat

rix

Mu

ltip

licat

ion

(AB

)C�

A(B

C)

Ass

oci

ativ

e P

rop

erty

of

Sca

lar

Mu

ltip

licat

ion

c(A

B)

�(c

A)B

�A

(cB

)

Lef

t D

istr

ibu

tive

Pro

per

tyC

(A�

B)

�C

A�

CB

Rig

ht

Dis

trib

uti

ve P

rop

erty

(A�

B)C

�A

C�

BC

Use

A�

,B�

,an

d C

�

to f

ind

eac

h p

rod

uct

.

a.(A

�B

)C

(A�

B)C

��

��

�

�

� �

b.

AC

�B

C

AC

�B

C�

� �

�

��

��

�

Not

e th

at a

lth

ough

th

e re

sult

s in

th

e ex

ampl

e il

lust

rate

th

e R

igh

t D

istr

ibu

tive

Pro

pert

y,th

ey d

o n

ot p

rove

it.

Use

A�

,B�

,C�

,an

d s

cala

r c

��

4 to

det

erm

ine

wh

eth

er

each

of

the

foll

owin

g eq

uat

ion

s is

tru

e fo

r th

e gi

ven

mat

rice

s.

1.c(

AB

) �

(cA

)Bye

s2.

AB

�B

An

o

3.B

C�

CB

no

4.(A

B)C

�A

(BC

)ye

s

5.C

(A�

B)

�A

C�

BC

no

6.c(

A�

B)

�cA

�cB

yes

��1 2�

�2

1

�3

64

21

32

5�

2

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�21

�5

�20

2�

4 �

13�

19

�14

�17

8�

1

2(

1) �

0(6)

2(�

2) �

0(3)

5

(1)

�(�

3)(6

)5(

�2)

�(�

3)(3

)4

(1)

�( �

3)(6

)4(

�2)

�( �

3)(3

)

2(1)

�1(

6)2(

�2)

�1(

3)

1�

2 6

32

0 5

�3

1�

2 6

34

�3

21

�12

�21

�5

�20

6(1

) �

( �3)

(6)

6(�

2) �

( �3)

(3)

7(1

) �

(�2)

(6)

7(�

2) �

(�2)

(3)

1�

2 6

36

�3

7�

2

1�

2 6

32

0 5

�3

4�

3 2

1

1�

2 6

32

0 5

�3

4�

3 2

1

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Mu

ltip

lyin

g M

atri

ces

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-3

4-3

Exam

ple

Exam

ple

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

Mu

ltip

lyin

g M

atri

ces

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-3

4-3

©G

lenc

oe/M

cGra

w-H

ill18

3G

lenc

oe A

lgeb

ra 2

Lesson 4-3

Det

erm

ine

wh

eth

er e

ach

mat

rix

pro

du

ct i

s d

efin

ed.I

f so

,sta

te t

he

dim

ensi

ons

ofth

e p

rod

uct

.

1.A

2 �

5�

B5

�1

2 �

12.

M1

�3

�N

3 �

21

�2

3.B

3 �

2�

A3

�2

un

def

ined

4.R

4 �

4�

S4

�1

4 �

1

5.X

3 �

3�

Y3

�4

3 �

46.

A6

�4

�B

4 �

56

�5

Fin

d e

ach

pro

du

ct,i

f p

ossi

ble

.

7.[3

2] �

[8]

8.�

9.�

10.

�n

ot

po

ssib

le

11.[

�3

4] �

[811

]12

.�

[2�

3�

2]

13.

�n

ot

po

ssib

le14

.�

15.

�16

.�

Use

A�

,B�

,C�

,a

nd

sca

lar

c�

2 to

det

erm

ine

wh

eth

er t

he

foll

owin

g eq

uat

ion

s ar

e tr

ue

for

the

give

n m

atri

ces.

17.(

AC

)c�

A(C

c)ye

s18

.AB

�B

An

o

19.B

(A�

C)

�A

B�

BC

no

20.(

A�

B)c

�A

c�

Bc

yes

3�

1 1

0�

32

5

12

1 2

1

4

4

2

2

2

01

1 1

10

�12

20

�

68

6

03

�3

02

�4

4 �

21

2

3

�6

61

512

3�

90

3 3

0

2�

2

45

�3

14

8

5

6 �

3

�2

32

6

�9

�6

�1

3

0�

1 2

2

1

3 �

11

3

�2

�3

�5

3

�2

1

3 �

11

28

�19

7�

92

�5

31

56

21

2

1

©G

lenc

oe/M

cGra

w-H

ill18

4G

lenc

oe A

lgeb

ra 2

Det

erm

ine

wh

eth

er e

ach

mat

rix

pro

du

ct i

s d

efin

ed.I

f so

,sta

te t

he

dim

ensi

ons

ofth

e p

rod

uct

.

1.A

7 �

4�

B4

�3

7 �

32.

A3

�5

�M

5 �

83

�8

3.M

2 �

1�

A1

�6

2 �

6

4.M

3 �

2�

A3

�2

un

def

ined

5.P

1 �

9�

Q9

�1

1 �

16.

P9

�1

�Q

1 �

99

�9

Fin

d e

ach

pro

du

ct,i

f p

ossi

ble

.

7.�

8.�

9.�

10.

�n

ot

po

ssib

le

11.[

40

2] �

[2]

12.

�[4

02]

13.

�14

.[�

15�

9] �

[�29

7�

75]

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,B�

,C�

,an

d s

cala

r c

�3

to d

eter

min

e w

het

her

th

e fo

llow

ing

equ

atio

ns

are

tru

e fo

r th

e gi

ven

mat

rice

s.

15.A

C�

CA

yes

16.A

(B�

C)

�B

A�

CA

no

17.(

AB

)c�

c(A

B)

yes

18.(

A�

C)B

�B

(A�

C)

no

REN

TALS

For

Exe

rcis

es 1

9–21

,use

th

e fo

llow

ing

info

rmat

ion

.F

or t

hei

r on

e-w

eek

vaca

tion

,th

e M

onto

yas

can

ren

t a

2-be

droo

m c

ondo

min

ium

for

$179

6,a

3-be

droo

m c

ondo

min

ium

for

$2

165,

or a

4-b

edro

om c

ondo

min

ium

for

$253

8.T

he

tabl

e sh

ows

the

nu

mbe

r of

u

nit

s in

eac

h o

f th

ree

com

plex

es.

19.W

rite

a m

atri

x th

at r

epre

sen

ts t

he

nu

mbe

r of

eac

h t

ype

of u

nit

ava

ilab

le a

t ea

ch c

ompl

ex a

nd

a m

atri

x th

at

,re

pres

ents

th

e w

eekl

y ch

arge

for

eac

h t

ype

of u

nit

.

20.I

f al

l of

th

e u

nit

s in

th

e th

ree

com

plex

es a

re r

ente

d fo

r th

e w

eek

at t

he

rate

s gi

ven

th

e M

onto

yas,

expr

ess

the

inco

me

of e

ach

of

the

thre

e co

mpl

exes

as

a m

atri

x.

21.W

hat

is

the

tota

l in

com

e of

all

th

ree

com

plex

es f

or t

he

wee

k?$5

26,0

54

172

,452

2

27,9

60

125

,642

$17

96

$21

65

$25

38

36

2422

2

932

42

18

2218

2-B

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om

3-B

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om

4-B

edro

om

Su

n H

aven

3624

22

Su

rfsi

de

2932

42

Sea

bre

eze

1822

18

�1

0

0�

1

40

�2

�1

13

31

6

11

23

�10

�

3010

15�

55

0 0

5�

62

3

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4

02

12

06

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2

1

3

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1

3

�1

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27

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93

24

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5

2

20

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6

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Pra

ctic

e (

Ave

rag

e)

Mu

ltip

lyin

g M

atri

ces

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-3

4-3



Readin

g t

o L

earn

Math

em

ati

csM

ult

iply

ing

Mat

rice

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-3

4-3

©G

lenc

oe/M

cGra

w-H

ill18

5G

lenc

oe A

lgeb

ra 2

Lesson 4-3

Pre-

Act

ivit

yH

ow c

an m

atri

ces

be

use

d i

n s

por

ts s

tati

stic

s?

Rea

d th

e in

trod

uct

ion

to

Les

son

4-3

at

the

top

of p

age

167

in y

our

text

book

.

Wri

te a

su

m t

hat

sh

ows

the

tota

l po

ints

sco

red

by t

he

Oak

lan

d R

aide

rsdu

rin

g th

e 20

00 s

easo

n.(

Th

e su

m w

ill

incl

ude

mu

ltip

lica

tion

s.D

o n

otac

tual

ly c

alcu

late

th

is s

um

.)

6 �

58 �

1 �

56 �

3 �

23 �

2 �

1 �

2 �

2

Rea

din

g t

he

Less

on

1.D

eter

min

e w

het

her

eac

h i

ndi

cate

d m

atri

x pr

odu

ct i

s de

fin

ed.I

f so

,sta

te t

he

dim

ensi

ons

of t

he

prod

uct

.If

not

,wri

te u

nd

efin

ed.

a.M

3 �

2an

d N

2 �

3M

N:

NM

:

b.

M1

�2

and

N1

�2

MN

:N

M:

c.M

4 �

1an

d N

1 �

4M

N:

NM

:

d.

M3

�4

and

N4

�4

MN

:N

M:

2.T

he

regi

onal

sal

es m

anag

er f

or a

ch

ain

of

com

pute

r st

ores

wan

ts t

o co

mpa

re t

he

reve

nu

efr

om s

ales

of

one

mod

el o

f n

oteb

ook

com

pute

r an

d on

e m

odel

of

prin

ter

for

thre

e st

ores

in h

is a

rea.

Th

e n

oteb

ook

com

pute

r se

lls

for

$185

0 an

d th

e pr

inte

r fo

r $1

75.T

he

nu

mbe

rof

com

pute

rs a

nd

prin

ters

sol

d at

th

e th

ree

stor

es d

uri

ng

Sep

tem

ber

are

show

n i

n t

he

foll

owin

g ta

ble.

Wri

te a

mat

rix

prod

uct

th

at t

he

man

ager

cou

ld u

se t

o fi

nd

the

tota

l re

ven

ue

for

com

pute

rs a

nd

prin

ters

for

eac

h o

f th

e th

ree

stor

es.(

Do

not

cal

cula

te t

he

prod

uct

.)

�

Hel

pin

g Y

ou

Rem

emb

er

3.M

any

stu

den

ts f

ind

the

proc

edu

re o

f m

atri

x m

ult

ipli

cati

on c

onfu

sin

g at

fir

st b

ecau

se i

tis

un

fam

ilia

r.T

hin

k of

an

eas

y w

ay t

o u

se t

he

lett

ers

R a

nd

C t

o re

mem

ber

how

to

mu

ltip

ly m

atri

ces

and

wh

at t

he

dim

ensi

ons

of t

he

prod

uct

wil

l be

.S

amp

le a

nsw

er:

Just

rem

emb

er R

C f

or

“ro

w,c

olu

mn

.”M

ult

iply

eac

h r

owo

f th

e fi

rst

mat

rix

by e

ach

co

lum

no

f th

e se

con

d m

atri

x.T

he

dim

ensi

on

s o

f th

e p

rod

uct

are

the

nu

mb

er o

f ro

ws

of

the

firs

t m

atri

x an

d t

he

nu

mb

er o

f co

lum

ns

of

the

seco

nd

mat

rix.1

850

17

51

2810

12

0516

6

9773

Sto

reC

om

pu

ters

Pri

nte

rs

A12

810

1

B20

516

6

C97

73

un

def

ined

3 �

4

1 �

14

�4

un

def

ined

un

def

ined

2 �

23

�3

©G

lenc

oe/M

cGra

w-H

ill18

6G

lenc

oe A

lgeb

ra 2

Fo

urt

h-O

rder

Det

erm

inan

tsT

o fi

nd

the

valu

e of

a 4

�4

dete

rmin

ant,

use

a m

eth

od c

alle

d ex

pan

sion

by

min

ors.

Fir

st w

rite

th

e ex

pan

sion

.Use

th

e fi

rst

row

of

the

dete

rmin

ant.

Rem

embe

r th

at t

he

sign

s of

th

e te

rms

alte

rnat

e.

��

��

��

��

�T

hen

eva

luat

e ea

ch 3

�3

dete

rmin

ant.

Use

an

y ro

w.

��

��

��

��

��

�2(

�16

�10

) �

�3(

24)

�5(

�6)

��

52�

�10

2

��

��

��

��

��

�6(

�16

�10

)�

�4(

�6)

�3(

�12

)�

�15

6�

�12

Fin

ally

,eva

luat

e th

e or

igin

al 4

�4

dete

rmin

ant.

��

Eva

luat

e ea

ch d

eter

min

ant.

1.

��

2.

��

3.

��

�10

9�

72�

676

14

30

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�3

64

51

12

42

5�

1

33

33

21

21

43

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52

50

1

12

31

43

�1

02

�5

44

1�

20

2�6(

�52

) �

3(�

102)

�2(

�15

6) �

7(�

12)

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846

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32

70

43

50

21

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30

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01

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21

60

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00

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50

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35

21

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20

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32

70

43

50

21

�4

60

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0

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-3

4-3


An

swer

s


Stu

dy G

uid

e a

nd I

nte

rven

tion

Tran

sfo

rmat

ion

s w

ith

Mat

rice

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-4

4-4

©G

lenc

oe/M

cGra

w-H

ill18

7G

lenc

oe A

lgeb

ra 2

Lesson 4-4

Tran

slat

ion

s an

d D

ilati

on

sM

atri

ces

that

rep

rese

nt

coor

din

ates

of

poin

ts o

n a

pla

ne

are

use

ful

in d

escr

ibin

g tr

ansf

orm

atio

ns.

Tran

slat

ion

a tr

ansf

orm

atio

n th

at m

oves

a f

igur

e fr

om o

ne lo

catio

n to

ano

ther

on

the

coor

dina

te p

lane

You

can

use

mat

rix

addi

tion

an

d a

tran

slat

ion

mat

rix

to f

ind

the

coor

din

ates

of

the

tran

slat

ed f

igu

re.

Dila

tio

na

tran

sfor

mat

ion

in w

hich

a f

igur

e is

enl

arge

d or

red

uced

You

can

use

sca

lar

mu

ltip

lica

tion

to

perf

orm

dil

atio

ns.

Fin

d t

he

coor

din

ates

of

the

vert

ices

of

the

imag

e of

�A

BC

wit

h v

erti

ces

A(�

5,4)

,B(�

1,5)

,an

d

C(�

3,�

1) i

f it

is

mov

ed 6

un

its

to t

he

righ

t an

d 4

un

its

dow

n.T

hen

gra

ph

�A

BC

and

its

im

age

�A

�B�C

�.

Wri

te t

he

vert

ex m

atri

x fo

r �

AB

C.

Add

th

e tr

ansl

atio

n m

atri

x to

th

e ve

rtex

m

atri

x of

�A

BC

. �

�

Th

e co

ordi

nat

es o

f th

e ve

rtic

es o

f �

A�B

�C�

are

A�(

1,0)

,B�(

5,1)

,an

d C

�(3,

�5)

.

For

Exe

rcis

es 1

an

d 2

use

th

e fo

llow

ing

info

rmat

ion

.Qu

adri

late

ral

QU

AD

wit

hve

rtic

es Q

(�1,

�3)

,U(0

,0),

A(5

,�1)

,an

d D

(2,�

5) i

s tr

ansl

ated

3 u

nit

s to

th

e le

ftan

d 2

un

its

up

.

1.W

rite

th

e tr

ansl

atio

n m

atri

x.

2.F

ind

the

coor

din

ates

of

the

vert

ices

of

Q�U

�A�D

�.Q

�(�

4,�

1),U

�,(�

3,2)

,A�(

2,1)

,D�(

�1,

�3)

For

Exe

rcis

es 3

–5,u

se t

he

foll

owin

g in

form

atio

n.T

he

vert

ices

of

�A

BC

are

A(4

,�2)

,B(2

,8),

and

C(8

,2).

Th

e tr

ian

gle

is d

ilat

ed s

o th

at i

ts p

erim

eter

is

one-

fou

rth

th

e or

igin

al p

erim

eter

.

3.W

rite

th

e co

ordi

nat

es o

f th

e ve

rtic

es o

f �

AB

Cin

a v

erte

x m

atri

x.

4.F

ind

the

coor

din

ates

of

the

vert

ices

of

imag

e �

A�B

�C�.

A� �1

,��,B

� �,2

�,C� �2

,�

5.G

raph

th

e pr

eim

age

and

the

imag

e.

1 � 21 � 2

1 � 24

28

�2

82

B�

C�

A

B

C

A�

x

y

O

�3

�3

�3

�3

2

22

2

15

3 0

1�

5

66

6 �

4�

4�

4�

5�

1�

3

45

�1

6

66

�4

�4

�4

�5

�1

�3

4

5�

1x

y

OA

�B

�

C�

AB

C

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill18

8G

lenc

oe A

lgeb

ra 2

Ref

lect

ion

s an

d R

ota

tio

ns

Ref

lect

ion

For

a r

efle

ctio

n ov

er t

he:

x-ax

isy-

axis

line

y�

x

Mat

rice

sm

ultip

ly t

he v

erte

x m

atrix

on

the

left

by:

Ro

tati

on

For

a c

ount

ercl

ockw

ise

rota

tion

abou

t th

e90

°18

0°27

0°

Mat

rice

s

orig

in o

f:

mul

tiply

the

ver

tex

mat

rix o

n th

e le

ft by

:

Fin

d t

he

coor

din

ates

of

the

vert

ices

of

the

imag

e of

�A

BC

wit

h

A(3

,5),

B(�

2,4)

,an

d C

(1,�

1) a

fter

a r

efle

ctio

n o

ver

the

lin

e y

�x.

Wri

te t

he

orde

red

pair

s as

a v

erte

x m

atri

x.T

hen

mu

ltip

ly t

he

vert

ex m

atri

x by

th

ere

flec

tion

mat

rix

for

y�

x.

� �

Th

e co

ordi

nat

es o

f th

e ve

rtic

es o

f A

�B�C

�ar

e A

�(5,

3),B

�(4,

�2)

,an

d C

�(�

1,1)

.

1.T

he

coor

din

ates

of

the

vert

ices

of

quad

rila

tera

l A

BC

Dar

e A

(�2,

1),B

(�1,

3),C

(2,2

),an

dD

(2,�

1).W

hat

are

th

e co

ordi

nat

es o

f th

e ve

rtic

es o

f th

e im

age

A�B

�C�D

�af

ter

are

flec

tion

ove

r th

e y-

axis

?A

�(2,

1),B

�(1,

3),C

�(�

2,2)

,D�(

�2,

�1)

2.T

rian

gle

DE

Fw

ith

ver

tice

s D

(�2,

5),E

(1,4

),an

d F

(0,�

1) i

s ro

tate

d 90

°co

un

terc

lock

wis

e ab

out

the

orig

in.

a.W

rite

th

e co

ordi

nat

es o

f th

e tr

ian

gle

in a

ver

tex

mat

rix.

b.

Wri

te t

he

rota

tion

mat

rix

for

this

sit

uat

ion

.

c.F

ind

the

coor

din

ates

of

the

vert

ices

of

�D

�E�F

�.D

�(�

5,�

2),E

�(�

4,1)

,F�(

1,0)

d.

Gra

ph �

DE

Fan

d �

D�E

�F�.

D�E

�

F�

DE

Fx

y

O

0�

11

0

�2

10

5

4�

1

54

�1

3�

21

3�

21

54

�1

01

10

0

1�

10

�1

0

0�

10

�1

10

01

10

�1

0

01

10

0�

1

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Tran

sfo

rmat

ion

s w

ith

Mat

rice

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-4

4-4

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Tran

sfo

rmat

ion

s w

ith

Mat

rice

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-4

4-4

©G

lenc

oe/M

cGra

w-H

ill18

9G

lenc

oe A

lgeb

ra 2

Lesson 4-4

For

Exe

rcis

es 1

–3,u

se t

he

foll

owin

g in

form

atio

n.

Tri

angl

e A

BC

wit

h v

erti

ces

A(2

,3),

B(0

,4),

and

C(�

3,�

3) i

s tr

ansl

ated

3 u

nit

s ri

ght

and

1 u

nit

dow

n.

1.W

rite

th

e tr

ansl

atio

n m

atri

x.

2.F

ind

the

coor

din

ates

of

�A

�B�C

�.A

�(5,

2),B

�(3,

3),C

�(0,

�4)

3.G

raph

th

e pr

eim

age

and

the

imag

e.

For

Exe

rcis

es 4

–6,u

se t

he

foll

owin

g in

form

atio

n.

Th

e ve

rtic

es o

f �

RS

Tar

e R

(�3,

1),S

(2,�

1),a

nd

T(1

,3).

Th

e tr

ian

gle

is d

ilat

ed s

o th

at i

ts p

erim

eter

is

twic

e th

e or

igin

al

peri

met

er.

4.W

rite

th

e co

ordi

nat

es o

f �

RS

Tin

a

vert

ex m

atri

x.

5.F

ind

the

coor

din

ates

of

the

imag

e �

R�S

�T�.

R�(

�6,

2),S

�(4,

�2)

,T�(2

,6)

6.G

raph

�R

ST

and

�R

�S�T

�.

For

Exe

rcis

es 7

–10,

use

th

e fo

llow

ing

info

rmat

ion

.T

he

vert

ices

of

�D

EF

are

D(4

,0),

E(0

,�1)

,an

d F

(2,3

).T

he

tria

ngl

e is

ref

lect

ed o

ver

the

x-ax

is.

7.W

rite

th

e co

ordi

nat

es o

f �

DE

Fin

a

vert

ex m

atri

x.

8.W

rite

th

e re

flec

tion

mat

rix

for

this

sit

uat

ion

.

9.F

ind

the

coor

din

ates

of

�D

�E�F

�.D

�(4,

0),E

�(0,

1),F

�(2,�

3)

10.G

raph

�D

EF

and

�D

�E�F

�.

For

Exe

rcis

es 1

1–14

,use

th

e fo

llow

ing

info

rmat

ion

.T

rian

gle

XY

Zw

ith

ver

tice

s X

(1,�

3),Y

(�4,

1),a

nd

Z(�

2,5)

is

rota

ted

180º

cou

nte

rclo

ckw

ise

abou

t th

e or

igin

.

11.W

rite

th

e co

ordi

nat

es o

f th

e tr

ian

gle

in a

ve

rtex

mat

rix.

12.W

rite

th

e ro

tati

on m

atri

x fo

r th

is s

itu

atio

n.

13.F

ind

the

coor

din

ates

of

�X

�Y�Z

�.X

�(�

1,3)

,Y�(

4,�

1),Z

�(2,

�5)

14.G

raph

th

e pr

eim

age

and

the

imag

e.

�1

0

0�

1

1

�4

�2

�3

15

X�

Y�

Z�

X

Y

Z

x

y

O

10

0�

1D

�E

�

F�

DE

F

x

y

O

40

20

�1

3

�3

21

1

�1

3R

�

S�

T�

R

S

T

x

y

O

3

33

�1

�1

�1

B�

C�

AB

C

A� x

y

O

©G

lenc

oe/M

cGra

w-H

ill19

0G

lenc

oe A

lgeb

ra 2

For

Exe

rcis

es 1

–3,u

se t

he

foll

owin

g in

form

atio

n.

Qu

adri

late

ral

WX

YZ

wit

h v

erti

ces

W(�

3,2)

,X(�

2,4)

,Y(4

,1),

and

Z(3

,0)

is t

ran

slat

ed 1

un

it l

eft

and

3 u

nit

s do

wn

.

1.W

rite

th

e tr

ansl

atio

n m

atri

x.

2.F

ind

the

coor

din

ates

of

quad

rila

tera

l W

�X�Y

�Z�.

W�(�

4,�

1),X

�(�3,

1),Y

�(3,�

2),Z

�(2,�

3)3.

Gra

ph t

he

prei

mag

e an

d th

e im

age.

For

Exe

rcis

es 4

–6,u

se t

he

foll

owin

g in

form

atio

n.

Th

e ve

rtic

es o

f �

RS

Tar

e R

(6,2

),S

(3,�

3),a

nd

T(�

2,5)

.Th

e tr

ian

gle

is d

ilat

ed s

o th

at i

ts p

erim

eter

is

one

hal

f th

e or

igin

al p

erim

eter

.

4.W

rite

th

e co

ordi

nat

es o

f �

RS

Tin

a v

erte

x m

atri

x.

5.F

ind

the

coor

din

ates

of

the

imag

e �

R�S

�T�.

R�(3

,1),

S�(1

.5,�

1.5)

,T�(�

1,2.

5)6.

Gra

ph �

RS

Tan

d �

R�S

�T�.

For

Exe

rcis

es 7

–10,

use

th

e fo

llow

ing

info

rmat

ion

.T

he

vert

ices

of

quad

rila

tera

l A

BC

Dar

e A

(�3,

2),B

(0,3

),C

(4,�

4),

and

D(�

2,�

2).T

he

quad

rila

tera

l is

ref

lect

ed o

ver

the

y-ax

is.

7.W

rite

th

e co

ordi

nat

es o

f A

BC

Din

a

vert

ex m

atri

x.

8.W

rite

th

e re

flec

tion

mat

rix

for

this

sit

uat

ion

.

9.F

ind

the

coor

din

ates

of

A�B

�C�D

�.A

�(3,2

),B

�(0,3

),C

�(�4,

�4)

,D�(2

,�2)

10.G

raph

AB

CD

and

A�B

�C�D

�.

11.A

RC

HIT

ECTU

RE

Usi

ng

arch

itec

tura

l de

sign

sof

twar

e,th

e B

radl

eys

plot

th

eir

kitc

hen

plan

s on

a g

rid

wit

h e

ach

un

it r

epre

sen

tin

g 1

foot

.Th

ey p

lace

th

e co

rner

s of

an

isl

and

at(2

,8),

(8,1

1),(

3,5)

,an

d (9

,8).

If t

he

Bra

dley

s w

ish

to

mov

e th

e is

lan

d 1.

5 fe

et t

o th

eri

ght

and

2 fe

et d

own

,wh

at w

ill

the

new

coo

rdin

ates

of

its

corn

ers

be?

(3.5

,6),

(9.5

,9),

(4.5

,3),

and

(10

.5,6

)

12.B

USI

NES

ST

he

desi

gn o

f a

busi

nes

s lo

go c

alls

for

loc

atin

g th

e ve

rtic

es o

f a

tria

ngl

e at

(1.5

,5),

(4,1

),an

d (1

,0)

on a

gri

d.If

des

ign

ch

ange

s re

quir

e ro

tati

ng

the

tria

ngl

e 90

ºco

un

terc

lock

wis

e,w

hat

wil

l th

e n

ew c

oord

inat

es o

f th

e ve

rtic

es b

e?(�

5,1.

5),(

�1,

4),a

nd

(0,

1)

�1

0

01

A�

B�

C�

D�

AB

C

D

x

y

O

�3

04

�2

2

3�

4�

2

63

�2

2�

35

R�

S�

T�

R

S

T

x

y

O

�1

�1

�1

�1

�3

�3

�3

�3

W�X

�

Y�

Z�

W

X

Y

Zx

y

O

Pra

ctic

e (

Ave

rag

e)

Tran

sfo

rmat

ion

s w

ith

Mat

rice

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-4

4-4


An

swer

s


Readin

g t

o L

earn

Math

em

ati

csTr

ansf

orm

atio

ns

wit

h M

atri

ces

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-4

4-4

©G

lenc

oe/M

cGra

w-H

ill19

1G

lenc

oe A

lgeb

ra 2

Lesson 4-4

Pre-

Act

ivit

yH

ow a

re t

ran

sfor

mat

ion

s u

sed

in

com

pu

ter

anim

atio

n?

Rea

d th

e in

trod

uct

ion

to

Les

son

4-4

at

the

top

of p

age

175

in y

our

text

book

.

Des

crib

e h

ow y

ou c

an c

han

ge t

he

orie

nta

tion

of

a fi

gure

wit

hou

t ch

angi

ng

its

size

or

shap

e.

Flip

(o

r re

flec

t) t

he

fig

ure

ove

r a

line.

Rea

din

g t

he

Less

on

1.a.

Wri

te t

he

vert

ex m

atri

x fo

r th

e qu

adri

late

ral

AB

CD

show

n i

n

the

grap

h a

t th

e ri

ght.

b.

Wri

te t

he v

erte

x m

atri

x th

at r

epre

sent

s th

e po

siti

on o

f th

e qu

adri

late

ral

A�B

�C�D

�th

atre

sult

s w

hen

qu

adri

late

ral

AB

CD

is t

ran

slat

ed 3

un

its

to t

he

righ

t an

d 2

un

its

dow

n.

2.D

escr

ibe

the

tran

sfor

mat

ion

th

at c

orre

spon

ds t

o ea

ch o

f th

e fo

llow

ing

mat

rice

s.

a.b

.

cou

nte

rclo

ckw

ise

rota

tio

n

tran

slat

ion

4 u

nit

s d

ow

n a

nd

ab

ou

t th

e o

rig

in o

f 18

0�3

un

its

to t

he

rig

ht

c.d

.

refl

ecti

on

ove

r th

e y-

axis

refl

ecti

on

ove

r th

e lin

e y

�x

Hel

pin

g Y

ou

Rem

emb

er

3.D

escr

ibe

a w

ay t

o re

mem

ber

wh

ich

of

the

refl

ecti

on m

atri

ces

corr

espo

nds

to

refl

ecti

onov

er t

he

x-ax

is.

Sam

ple

an

swer

:Th

e o

nly

ele

men

ts u

sed

in t

he

refl

ecti

on

mat

rice

s ar

e 0,

1,an

d �

1.F

or

such

a 2

�2

mat

rix

Mto

hav

e th

e p

rop

erty

th

at

M�

�,t

he

elem

ents

in t

he

top

ro

w m

ust

be

1 an

d 0

(in

th

at

ord

er),

and

ele

men

ts in

th

e b

ott

om

ro

w m

ust

be

0 an

d �

1 (i

n t

hat

ord

er).

x

�y

x

y

01

10

�1

0

01

3

33

�4

�4

�4

�1

0

0�

1

�1

54

1�

11

�6

�5

�4

21

�2

1

3�

4�

3x

y

O

A

B

CD

©G

lenc

oe/M

cGra

w-H

ill19

2G

lenc

oe A

lgeb

ra 2

Pro

per

ties

of

Det

erm

inan

tsT

he fo

llow

ing

prop

erti

es o

ften

hel

p w

hen

eval

uati

ng d

eter

min

ants

.

•If

all

th

e el

emen

ts o

f a

row

(or

col

um

n)

are

zero

,th

e va

lue

of t

he

dete

rmin

ant

is z

ero.

�0

(a�

0) �

(0 �

b)

�0

•M

ult

iply

ing

all

the

elem

ents

of

a ro

w (

or c

olu

mn

) by

a c

onst

ant

is e

quiv

alen

t to

mu

ltip

lyin

g th

e va

lue

of t

he

dete

rmin

ant

by t

he

con

stan

t.

3�

3[4(

3) �

5(�

1)]

�51

3[12

�5]

�51

12(3

) �

5(�

3)�

51

•If

tw

o ro

ws

(or

colu

mn

s) h

ave

equ

al c

orre

spon

din

g el

emen

ts,t

he

valu

e of

th

ede

term

inan

t is

zer

o.

�0

5(�

3) �

(�3)

(5)

�0

•T

he

valu

e of

a d

eter

min

ant

is u

nch

ange

d if

an

y m

ult

iple

of

a ro

w (

or c

olu

mn

) is

add

ed t

o co

rres

pon

din

g el

emen

ts o

f an

oth

er r

ow (

or c

olu

mn

).

�4(

5) �

2(�

3)�

6(5)

�2(

2)�

2620

�6

�26

30 �

4�

26(R

ow 2

is a

dded

to

row

1.)

•If

tw

o ro

ws

(or

colu

mn

s) a

re i

nte

rch

ange

d,th

e si

gn o

f th

e de

term

inan

t is

ch

ange

d.

�4(

8) �

(�3)

(5)

��

[(�

3)(5

) �

4(8)

]�

4732

�15

�47

�[�

15 �

32]

�47

•T

he

valu

e of

th

e de

term

inan

t is

un

chan

ged

if r

ow 1

is

inte

rch

ange

d w

ith

col

um

n 1

,an

d ro

w 2

is

inte

rch

ange

d w

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tabl

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1815

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En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

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____

____

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AT

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____

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ER

IOD

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4-5

4-5


An

swer

s


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dy G

uid

e a

nd I

nte

rven

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mer

’s R

ule

NA

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____

____

____

____

____

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____

____

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AT

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4-6

4-6

©G

lenc

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cGra

w-H

ill19

9G

lenc

oe A

lgeb

ra 2

Lesson 4-6

Syst

ems

of

Two

Lin

ear

Equ

atio

ns

Det

erm

inan

ts p

rovi

de a

way

for

sol

vin

g sy

stem

sof

equ

atio

ns.

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mer

’s R

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fo

r

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sol

utio

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line

ar s

yste

m o

f eq

uatio

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iab

le S

yste

ms

cx�

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f

is (

x, y

) w

here

x�

,

y�

, an

d �

0.

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mer

’s R

ule

to

solv

e th

e sy

stem

of

equ

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10y

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10x

�25

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mer

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luat

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ch d

eter

min

ant.

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impl

ify.

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or �

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e so

luti

on i

s �

,��.

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mer

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ch s

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f eq

uat

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s.

1.3x

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Syst

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of

Thre

e Li

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uat

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sys

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quat

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l

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re x

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d z

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�0.

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stan

ts f

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th

e eq

uat

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s to

for

m t

he

dete

rmin

ants

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enev

alu

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each

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t.

x�

y

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�or

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or �

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e so

luti

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4-6

4-6

Exam

ple

Exam

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Exer

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Skil

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4-6

4-6

©G

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1G

lenc

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lgeb

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Lesson 4-6

Use

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’s R

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to

solv

e ea

ch s

yste

m o

f eq

uat

ion

s.

1.2a

�3b

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2.3x

�y

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4)2x

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17.G

EOM

ETRY

Th

e tw

o si

des

of a

n a

ngl

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e co

nta

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th

e li

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es o

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rtex

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angl

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lenc

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Use

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solv

e ea

ch s

yste

m o

f eq

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ion

s.

1.2x

�y

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EOM

ETRY

Th

e tw

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des

of a

n a

ngl

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nta

ined

in

th

e li

nes

wh

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ns

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x�

y�

6 an

d 2x

�y

�1.

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d th

e co

ordi

nat

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f th

e ve

rtex

of

the

angl

e.(2

,�3)

17.G

EOM

ETRY

Tw

o si

des

of a

par

alle

logr

am a

re c

onta

ined

in

th

e li

nes

wh

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ns

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5y�

1 an

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02x

�0.

3y�

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9.F

ind

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4)

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18.x

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24.L

AN

DSC

API

NG

A m

emor

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gard

en b

ein

g pl

ante

d in

fr

ont

of a

mu

nic

ipal

lib

rary

wil

l co

nta

in t

hre

e ci

rcu

lar

beds

th

at a

re t

ange

nt

to e

ach

oth

er.A

lan

dsca

pe a

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itec

t h

as p

repa

red

a sk

etch

of

the

desi

gn f

or t

he

gard

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sin

g C

AD

(co

mpu

ter-

aide

d dr

afti

ng)

sof

twar

e,as

sh

own

at

the

righ

t.T

he

cen

ters

of

the

thre

e ci

rcu

lar

beds

are

rep

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nte

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poi

nts

A,B

,an

d C

.Th

e di

stan

ce f

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Bis

15

feet

,th

e di

stan

ce f

rom

Bto

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13

feet

,an

d th

e di

stan

ce f

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A

to C

is 1

6 fe

et.W

hat

is

the

radi

us

of e

ach

of

the

circ

ula

rbe

ds?

circ

le A

:9

ft,c

ircl

e B

:6

ft,c

ircl

e C

:7

ft

A

BC

13 ft

16 ft

15 ft

4 � 3

4 � 31 � 3

Pra

ctic

e (

Ave

rag

e)

Cra

mer

’s R

ule

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-6

4-6


An

swer

s


Readin

g t

o L

earn

Math

em

ati

csC

ram

er’s

Ru

le

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

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____

____

__P

ER

IOD

____

_

4-6

4-6

©G

lenc

oe/M

cGra

w-H

ill20

3G

lenc

oe A

lgeb

ra 2

Lesson 4-6

Pre-

Act

ivit

yH

ow i

s C

ram

er’s

Ru

le u

sed

to

solv

e sy

stem

s of

eq

uat

ion

s?

Rea

d th

e in

trod

uct

ion

to

Les

son

4-6

at

the

top

of p

age

189

in y

our

text

book

.

A t

rian

gle

is b

oun

ded

by t

he

x-ax

is,t

he

lin

e y

�x,

and

the

lin

e

y�

�2x

�10

.Wri

te t

hre

e sy

stem

s of

equ

atio

ns

that

you

cou

ld u

se t

o fi

nd

the

thre

e ve

rtic

es o

f th

e tr

ian

gle.

(Do

not

act

ual

ly f

ind

the

vert

ices

.)

x�

0,y

�x;

x�

0,y

��

2x�

10;

y�

x,y

��

2x�

10

Rea

din

g t

he

Less

on

1.S

upp

ose

that

you

are

ask

ed t

o so

lve

the

foll

owin

g sy

stem

of

equ

atio

ns

by C

ram

er’s

Ru

le.

3x�

2y�

72x

�3y

�22

Wit

hou

t ac

tual

ly e

valu

atin

g an

y de

term

inan

ts,i

ndi

cate

wh

ich

of

the

foll

owin

g ra

tios

of

dete

rmin

ants

giv

es t

he

corr

ect

valu

e fo

r x.

B

A.

B.

C.

2.In

you

r te

xtbo

ok,t

he

stat

emen

ts o

f C

ram

er’s

Ru

le f

or t

wo

vari

able

s an

d th

ree

vari

able

ssp

ecif

y th

at t

he

dete

rmin

ant

form

ed f

rom

th

e co

effi

cien

ts o

f th

e va

riab

les

can

not

be

0.If

th

e de

term

inan

t is

zer

o,w

hat

do

you

kn

ow a

bou

t th

e sy

stem

an

d it

s so

luti

ons?

Th

e sy

stem

co

uld

be

a d

epen

den

t sy

stem

an

d h

ave

infi

nit

ely

man

yso

luti

on

s,o

r it

co

uld

be

an in

con

sist

ent

syst

em a

nd

hav

e n

o s

olu

tio

ns.

Hel

pin

g Y

ou

Rem

emb

er

3.S

ome

stud

ents

hav

e tr

oubl

e re

mem

beri

ng h

ow t

o ar

rang

e th

e de

term

inan

ts t

hat

are

used

in s

olvi

ng

a sy

stem

of

two

lin

ear

equ

atio

ns

by C

ram

er’s

Ru

le.W

hat

is

a go

od w

ay t

ore

mem

ber

this

?

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ple

an

swer

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Db

e th

e d

eter

min

ant

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the

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.Let

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rmin

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ed b

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plac

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umn

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with

the

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ide

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the

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the

det

erm

inan

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rmed

by

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th

e se

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mn

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en

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Co

mm

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etw

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ram

at

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icat

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ote

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s.T

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ich

sig

nal

s ca

n b

e tr

ansm

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d re

ceiv

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y ea

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ter.

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gen

erat

e a

mat

rix

to d

escr

ibe

this

net

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k.

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rix

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a c

omm

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ose

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ge f

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on

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ter

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usi

ng

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tly

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oth

er

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pute

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elay

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an b

e sh

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th

at t

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atri

x A

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the

nu

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ne

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a

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d st

atio

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or e

xam

ple,

a m

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ay b

e se

nt

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sta

tion

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atio

n 5

by

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n t

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d t

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rite

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um

ber

of

way

s th

e m

essa

ges

can

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t fo

r ea

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3.

A:

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x A

2to

th

eco

mm

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icat

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s di

agra

m t

o ve

rify

th

at t

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corr

ect.

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rix

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sin

g ex

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11

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om1

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pute

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The

ent

ry in

pos

itio

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ents

the

num

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ays

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end

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com

pute

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ter

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rect

ly.C

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mat

rix

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th

e di

agra

m t

o ve

rify

th

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trie

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orex

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ere

is o

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ter

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3,4

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En

rich

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NA

ME

____

____

____

____

____

____

____

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AT

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ER

IOD

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4-6

4-6



Stu

dy G

uid

e a

nd I

nte

rven

tion

Iden

tity

an

d In

vers

e M

atri

ces

NA

ME

____

____

____

____

____

____

____

____

____

____

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AT

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____

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ER

IOD

____

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4-7

4-7

©G

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5G

lenc

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lgeb

ra 2

Lesson 4-7

Iden

tity

an

d In

vers

e M

atri

ces

Th

e id

enti

ty m

atri

x fo

r m

atri

x m

ult

ipli

cati

on i

s a

squ

are

mat

rix

wit

h 1

s fo

r ev

ery

elem

ent

of t

he

mai

n d

iago

nal

an

d ze

ros

else

wh

ere.

Iden

tity

Mat

rix

If A

is a

n n

�n

mat

rix a

nd I

is t

he id

entit

y m

atrix

,fo

r M

ult

iplic

atio

nth

en A

�I

�A

and

I�

A�

A.

If a

n n

�n

mat

rix

Ah

as a

n i

nve

rse

A�

1 ,th

en A

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erm

ine

wh

eth

er X

�

and

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e in

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em

atri

ces.

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d X

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.

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or

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or

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ce X

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are

inve

rse

mat

rice

s.

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ach

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r of

mat

rice

s ar

e in

vers

es.

1.an

d 2.

and

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d

yes

yes

no

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d 5.

and

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d

yes

no

yes

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d 8.

and

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d

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yes

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10.

and

11.

and

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and

no

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Fin

d In

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of

a 2

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rix

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inve

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mat

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ave

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inve

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of N

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st f

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valu

e of

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e de

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t.

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ce t

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rmin

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eck

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ach

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ists

.

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no

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ts

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6.

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rven

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onti

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)

Iden

tity

an

d In

vers

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atri

ces

NA

ME

____

____

____

____

____

____

____

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AT

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ER

IOD

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4-7

4-7

Exam

ple

Exam

ple

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

Iden

tity

an

d In

vers

e M

atri

ces

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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AT

E__

____

____

__P

ER

IOD

____

_

4-7

4-7

©G

lenc

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7G

lenc

oe A

lgeb

ra 2

Lesson 4-7

Det

erm

ine

wh

eth

er e

ach

pai

r of

mat

rice

s ar

e in

vers

es.

1.X

�,Y

�ye

s2.

P�

,Q�

yes

3.M

�,N

�n

o4.

A�

,B�

yes

5.V

�,W

�ye

s6.

X�

,Y�

yes

7.G

�,H

�ye

s8.

D�

,E�

no

Fin

d t

he

inve

rse

of e

ach

mat

rix,

if i

t ex

ists

.

9.�

10.

�

11.

no

inve

rse

exis

ts12

.

13.

14.

no

inve

rse

exis

ts

15.

16.

�

17.

�18

.n

o in

vers

e ex

ists

19.

�20

.2

00

21 � 4

20

02

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8�

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1010

1

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10

8 1

0�

8

10

8

54

07

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7 �

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2

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45

1

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11

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1

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1

3

6 �

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2

31

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11 � 6

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3

0

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21 � 24

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6

09

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2

2

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lenc

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8G

lenc

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lgeb

ra 2

Det

erm

ine

wh

eth

er e

ach

pai

r of

mat

rice

s ar

e in

vers

es.

1.M

�,N

�n

o2.

X�

,Y�

yes

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�,B

�ye

s4.

P�

,Q�

yes

Det

erm

ine

wh

eth

er e

ach

sta

tem

ent

is t

rue

or f

als

e.

5.A

ll s

quar

e m

atri

ces

hav

e m

ult

ipli

cati

ve i

nve

rses

.fa

lse

6.A

ll s

quar

e m

atri

ces

hav

e m

ult

ipli

cati

ve i

den

titi

es.

tru

e

Fin

d t

he

inve

rse

of e

ach

mat

rix,

if i

t ex

ists

.

7.8.

9.�

10.

11.

12.

no

inve

rse

exis

ts

GEO

MET

RYF

or E

xerc

ises

13–

16,u

se t

he

figu

re a

t th

e ri

ght.

13.W

rite

th

e ve

rtex

mat

rix

Afo

r th

e re

ctan

gle.

14.U

se m

atri

x m

ult

ipli

cati

on t

o fi

nd

BA

if B

�.

15.G

raph

th

e ve

rtic

es o

f th

e tr

ansf

orm

ed t

rian

gle

on t

he

prev

iou

s gr

aph

.D

escr

ibe

the

tran

sfor

mat

ion

.d

ilati

on

by

a sc

ale

fact

or

of

1.5

16.M

ake

a co

nje

ctu

re a

bou

t w

hat

tra

nsf

orm

atio

n B

�1

desc

ribe

s on

a c

oord

inat

e pl

ane.

dila

tio

n b

y a

scal

e fa

cto

r o

f

17.C

OD

ESU

se t

he

alph

abet

tab

le b

elow

an

d th

e in

vers

e of

cod

ing

mat

rix

C�

to

deco

de t

his

mes

sage

:19

|14|11

|13|11

|22|55

|65|57

|60|2

|1|52

|47|33

|51|56

|55.

CH

EC

K_Y

OU

R_A

NS

WE

RS

CO

DE

A1

B2

C3

D4

E5

F6

G7

H8

I9

J10

K11

L12

M13

N14

O15

P16

Q17

R18

S19

T20

U21

V22

W23

X24

Y25

Z26

–0

12

21

2 � 3

1.5

67.

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3

61.

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1.5

1.5

0

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)

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)

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( 1.5

, 3)

x

y

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( 2, –

1)

( 1, 2

)

( 4, 4

)

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)

46

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21 � 11

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53

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32Pra

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Ave

rag

e)

Iden

tity

an

d In

vers

e M

atri

ces

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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AT

E__

____

____

__P

ER

IOD

____

_

4-7

4-7



Readin

g t

o L

earn

Math

em

ati

csId

enti

ty a

nd

Inve

rse

Mat

rice

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-7

4-7

©G

lenc

oe/M

cGra

w-H

ill20

9G

lenc

oe A

lgeb

ra 2

Lesson 4-7

Pre-

Act

ivit

yH

ow a

re i

nve

rse

mat

rice

s u

sed

in

cry

pto

grap

hy?

Rea

d th

e in

trod

uct

ion

to

Les

son

4-7

at

the

top

of p

age

195

in y

our

text

book

.

Ref

er t

o th

e co

de t

able

giv

en i

n t

he

intr

odu

ctio

n t

o th

is l

esso

n.S

upp

ose

that

you

rec

eive

a m

essa

ge c

oded

by

this

sys

tem

as

foll

ows:

1612

51

195

25

1325

618

95

144.

Dec

ode

the

mes

sage

.P

leas

e b

e m

y fr

ien

d.

Rea

din

g t

he

Less

on

1.In

dica

te w

het

her

eac

h o

f th

e fo

llow

ing

stat

emen

ts i

s tr

ue

or f

alse

.

a.E

very

ele

men

t of

an

ide

nti

ty m

atri

x is

1.

fals

e

b.

Th

ere

is a

3 �

2 id

enti

ty m

atri

x.fa

lse

c.T

wo

mat

rice

s ar

e in

vers

es o

f ea

ch o

ther

if

thei

r pr

odu

ct i

s th

e id

enti

ty m

atri

x.tr

ue

d.

If M

is a

mat

rix,

M�

1re

pres

ents

th

e re

cipr

ocal

of

M.

fals

e

e.N

o 3

�2

mat

rix

has

an

in

vers

e.tr

ue

f.E

very

squ

are

mat

rix

has

an

in

vers

e.fa

lse

g.If

th

e tw

o co

lum

ns

of a

2 �

2 m

atri

x ar

e id

enti

cal,

the

mat

rix

does

not

hav

e an

inve

rse.

tru

e

2.E

xpla

in h

ow t

o fi

nd

the

inve

rse

of a

2 �

2 m

atri

x.D

o n

ot u

se a

ny

spec

ial

mat

hem

atic

alsy

mbo

ls i

n y

our

expl

anat

ion

.

Sam

ple

an

swer

:F

irst

fin

d t

he

det

erm

inan

t o

f th

e m

atri

x.If

it is

zer

o,t

hen

the

mat

rix

has

no

inve

rse.

If t

he

det

erm

inan

t is

no

t ze

ro,f

orm

a n

ewm

atri

x as

fo

llow

s.In

terc

han

ge

the

top

left

an

d b

ott

om

rig

ht

elem

ents

.C

han

ge

the

sig

ns

but

no

t th

e p

osi

tio

ns

of

the

oth

er t

wo

ele

men

ts.

Mu

ltip

ly t

he

resu

ltin

g m

atri

x by

th

e re

cip

roca

l of

the

det

erm

inan

t o

f th

eo

rig

inal

mat

rix.

Th

e re

sult

ing

mat

rix

is t

he

inve

rse

of

the

ori

gin

al m

atri

x.

Hel

pin

g Y

ou

Rem

emb

er

3.O

ne

way

to

rem

embe

r so

met

hin

g is

to

expl

ain

it

to a

not

her

per

son

.Su

ppos

e th

at y

ou a

rest

udy

ing

wit

h a

cla

ssm

ate

wh

o is

hav

ing

trou

ble

rem

embe

rin

g h

ow t

o fi

nd

the

inve

rse

ofa

2 �

2 m

atri

x.H

e re

mem

bers

how

to

mov

e el

emen

ts a

nd

chan

ge s

ign

s in

th

e m

atri

x,bu

t th

inks

th

at h

e sh

ould

mu

ltip

ly b

y th

e de

term

inan

t of

th

e or

igin

al m

atri

x.H

ow c

anyo

u h

elp

him

rem

embe

r th

at h

e m

ust

mu

ltip

ly b

y th

e re

cipr

ocal

of t

his

det

erm

inan

t?

Sam

ple

an

swer

:If

th

e d

eter

min

ant

of

the

mat

rix

is 0

,its

rec

ipro

cal i

su

nd

efin

ed.T

his

ag

rees

wit

h t

he

fact

th

at if

th

e d

eter

min

ant

of

a m

atri

x is

0,th

e m

atri

x d

oes

no

t h

ave

an in

vers

e.

©G

lenc

oe/M

cGra

w-H

ill21

0G

lenc

oe A

lgeb

ra 2

Per

mu

tati

on

Mat

rice

sA

per

mu

tati

on m

atri

x is

a s

quar

e m

atri

x in

wh

ich

ea

ch r

ow a

nd

each

col

um

n h

as o

ne

entr

y th

at i

s 1.

All

th

e ot

her

en

trie

s ar

e 0.

Fin

d th

e in

vers

e of

ape

rmu

tati

on m

atri

x in

terc

han

gin

g th

e ro

ws

and

colu

mn

s.F

or e

xam

ple,

row

1 i

s in

terc

han

ged

wit

hco

lum

n 1

,row

2 i

s in

terc

han

ged

wit

h c

olu

mn

2.

Pis

a 4

�4

perm

uta

tion

mat

rix.

P�

1is

th

e in

vers

e of

P.

Sol

ve e

ach

pro

ble

m.

1.T

her

e is

just

on

e 2

�2

perm

uta

tion

2.

Fin

d th

e in

vers

e of

th

e m

atri

x yo

u w

rote

m

atri

x th

at i

s n

ot a

lso

an i

den

tity

in

Exe

rcis

e 1.

Wh

at d

o yo

u n

otic

e?m

atri

x.W

rite

th

is m

atri

x.

Th

e tw

o m

atri

ces

are

the

sam

e.

3.S

how

th

at t

he

two

mat

rice

s in

Exe

rcis

es 1

an

d 2

are

inve

rses

.

�

4.W

rite

th

e in

vers

e of

th

is m

atri

x.

B�

B�

1�

5.U

se B

�1

from

pro

blem

4.V

erif

y th

at B

and

B�

1ar

e in

vers

es.

�

6.P

erm

uta

tion

mat

rice

s ca

n b

e u

sed

to w

rite

an

d de

ciph

er c

odes

.To

see

how

th

is i

s do

ne,

use

th

e m

essa

ge m

atri

x M

and

mat

rix

Bfr

om p

robl

em 4

.Fin

d m

atri

x C

so t

hat

Ceq

ual

s th

e pr

odu

ct M

B.U

se t

he

rule

s be

low

.

0 tim

es a

lette

r �

0

1 tim

es a

lette

r �

the

sam

e le

tter

0 pl

us a

lette

r �

the

sam

e le

tter

7.N

ow f

ind

the

prod

uct

CB

�1 .

Wh

at d

o yo

u n

otic

e?

�

�

Mu

ltip

lyin

g M

by B

enco

des

th

e m

essa

ge.

To d

ecip

her

,mu

ltip

ly b

y B

�1 .

SH

E

SA

W

HI

M

01

00

01

10

0

HE

S

AW

S

I

MH

01

00

01

10

0

0 �

0 �

0 �

0 �

1 �

10

�1

�0

�0

�1

�0

0 �

0 �

0 �

1 �

1 �

01

�0

�0

�0

�0

�1

1 �

1 �

0 �

0 �

0 �

01

�0

�0

�1

�0

�0

0 �

0 �

1 �

0 �

0 �

10

�1

�1

�0

�0

�0

0 �

0 �

1 �

1 �

0 �

0

01

00

01

10

0

00

11

00

01

0

10

01

0 �

1 �

1 �

10

�1

�1

�0

1 �

0 �

0 �

11

�1

�0

�0

01

10

01

10

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-7

4-7

P�

P�

1�

00

01

01

00

10

00

00

10

00

10

01

00

00

01

10

00

M�

C�

HE

S

AW

S

I

MH

SH

E

SA

W

HI

M


An

swer

s


Stu

dy G

uid

e a

nd I

nte

rven

tion

Usi

ng

Mat

rice

s to

So

lve

Sys

tem

s o

f E

qu

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-8

4-8

©G

lenc

oe/M

cGra

w-H

ill21

1G

lenc

oe A

lgeb

ra 2

Lesson 4-8

Wri

te M

atri

x Eq

uat

ion

sA

mat

rix

equ

atio

nfo

r a

syst

em o

f eq

uat

ion

s co

nsi

sts

of t

he

prod

uct

of

the

coef

fici

ent

and

vari

able

mat

rice

s on

th

e le

ft a

nd

the

con

stan

t m

atri

x on

th

eri

ght

of t

he

equ

als

sign

.

Wri

te a

mat

rix

equ

atio

n f

or e

ach

sys

tem

of

equ

atio

ns.

a.3x

�7y

�12

x�

5y�

�8

Det

erm

ine

the

coef

fici

ent,

vari

able

,an

dco

nst

ant

mat

rice

s.

��

12

�8

x

y

3�

7 1

5

b.

2x�

y�

3z�

�7

x�

3y�

4z�

157x

�2y

�z

��

28

� �

�

7

15

�28

x

y

z

2�

13

13

�4

72

1

Exam

ple

Exam

ple

Exer

cises

Exer

cises

Wri

te a

mat

rix

equ

atio

n f

or e

ach

sys

tem

of

equ

atio

ns.

1.2x

�y

�8

2.4x

�3y

�18

3.7x

�2y

�15

5x�

3y�

�12

x�

2y�

123x

�y

��

10

� �

�

�

� �

4.4x

�6y

�20

5.5x

�2y

�18

6.3x

�y

�24

3x�

y�

8�0

x�

�4y

�25

3y�

80 �

2x

� �

�

�

� �

7.2x

�y

�7z

�12

8.5x

�y

�7z

�32

5x�

y�

3z�

15x

�3y

�2z

��

18x

�2y

�6z

�25

2x�

4y�

3z�

12

� �

�

�

9.4x

�3y

�z

��

100

10.x

�3y

�7z

�27

2x�

y�

3z�

�64

2x�

y�

5z�

485x

�3y

�2z

�8

4x�

2y�

3z�

72

� �

�

�

11.2

x�

3y�

9z�

�10

812

.z�

45 �

3x�

2yx

�5z

�40

�2y

2x�

3y�

z�

603x

�5y

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�4z

x�

4y�

2z�

120

� �

�

�

45

60

1

20

x

y

z

3�

21

23

�1

1�

42

�10

8

40

89

x

y

z

23

�9

1�

25

35

�4

27

48

72

x

y

z

1�

37

21

�5

4�

23

�10

0

�64

8

x

y

z

4�

3�

12

1�

35

3�

2

32

�

18

12

x

y

z

5�

17

13

�2

24

�3

12

15

25

x

y

z

21

75

�1

31

2�

6

24

80

x

y

3�

12

31

82

5x

y

5

21

42

0�

8x

y

4

�6

31

15

�

10

x

y

7�

23

11

81

2x

y

4

�3

12

8

�12

x

y

2

15

�3

©G

lenc

oe/M

cGra

w-H

ill21

2G

lenc

oe A

lgeb

ra 2

Solv

e Sy

stem

s o

f Eq

uat

ion

sU

se i

nve

rse

mat

rice

s to

sol

ve s

yste

ms

of e

quat

ion

sw

ritt

en a

s m

atri

x eq

uat

ion

s.

So

lvin

g M

atri

x E

qu

atio

ns

If A

X�

B,

then

X�

A�

1 B,

whe

re A

is t

he c

oeffi

cien

t m

atrix

, X

is t

he v

aria

ble

mat

rix,

and

Bis

the

con

stan

t m

atrix

.

Sol

ve

�.

In t

he

mat

rix

equ

atio

n A

�,X

�,a

nd

B�

.

Ste

p 1

Fin

d th

e in

vers

e of

th

e co

effi

cien

t m

atri

x.

A�

1�

or

.

Ste

p 2

Mu

ltip

ly e

ach

sid

e of

th

e m

atri

x eq

uat

ion

by

the

inve

rse

mat

rix.

� �

��

Mul

tiply

eac

h si

de b

y A

�1 .

� �

Mul

tiply

mat

rices

.

�S

impl

ify.

Th

e so

luti

on i

s (2

,�2)

.

Sol

ve e

ach

mat

rix

equ

atio

n o

r sy

stem

of

equ

atio

ns

by

usi

ng

inve

rse

mat

rice

s.

1.�

�2.

� �

3.�

�

(5,�

3)n

o s

olu

tio

n(1

3,�

18)

4.�

�5.

� �

6.�

�

��,�

�(5

7,�

31)

��,

�7.

4x�

2y�

228.

2x�

y�

29.

3x�

4y�

126x

�4y

��

2x

�2y

�46

5x�

8y�

�8

(3,�

5)(1

0,18

)(3

2,�

21)

10.x

�3y

��

511

.5x

�4y

�5

12.3

x�

2y�

52x

�7y

�8

9x�

8y�

0x

�4y

�20

(�59

,18)

�,

�(�

2,�

5.5)

45 � 7610 � 19

15 � 79 � 7

3 � 21 � 4

3

�6

x

y

12

3�

1�

15

6

x

y

36

59

4

�8

x

y

2�

3 2

5

3

�7

x

y

32

54

16

12

x

y

�4

�8

6

12

�2

18

x

y

2

4 3

�1

2

�2

x

y

16

�

16

1 � 8x

y

1

0 0

1

6

4

4

�2

�6

51 � 8

x

y

52

64

4

�2

�6

51 � 8

4

�2

�6

51 � 8

4

�2

�6

51

� 20 �

12

6

4

x

y

52

64

6

4

x

y

52

64

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Usi

ng

Mat

rice

s to

So

lve

Sys

tem

s o

f E

qu

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-8

4-8

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Usi

ng

Mat

rice

s to

So

lve

Sys

tem

s o

f E

qu

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-8

4-8

©G

lenc

oe/M

cGra

w-H

ill21

3G

lenc

oe A

lgeb

ra 2

Lesson 4-8

Wri

te a

mat

rix

equ

atio

n f

or e

ach

sys

tem

of

equ

atio

ns.

1.x

�y

�5

2.3a

�8b

�16

2x�

y�

14a

�3b

�3

� �

�

�

3.m

�3n

��

34.

2c�

3d�

64m

�3n

�6

3c�

4d�

7

� �

�

�

5.r

�s

�1

6.x

�y

�5

2r�

3s�

123x

�2y

�10

� �

�

�

7.6x

�y

�2z

��

48.

a�

b�

c�

5�

3x�

2y�

z�

103a

�2b

�c

�0

x�

y�

z�

32a

�3b

�8

� �

�

�

Sol

ve e

ach

mat

rix

equ

atio

n o

r sy

stem

of

equ

atio

ns

by

usi

ng

inve

rse

mat

rice

s.

9.�

�(2

,�3)

10.

��

(3,�

2)

11.

��

(3,�

2)12

.�

�(3

,2)

13.

��

�7,�

14.

��

�1,�

15.p

�3q

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16.�

x�

3y�

22p

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6(0

,�2)

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(1,�

1)

17.2

m�

2n�

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18.�

3a�

b�

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6m�

4n�

�18

(�1,

�3)

5a�

2b�

14(4

,3)

5 � 31

5

2m

n

56

12

�6

1 � 32

5 1

2c

d

3

12

2�

6

15

23

m

n

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3 5

4�

1

7a

b

5

8 3

1

6

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x

y

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13

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w

z

13

43

5

0

8

a

b

c

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11

32

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23

0

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10

3

x

y

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32

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1

11

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10

x

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11

32

1

12

r

s

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12

3

6

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c

d

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6m

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a

b

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1

x

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2�

1

©G

lenc

oe/M

cGra

w-H

ill21

4G

lenc

oe A

lgeb

ra 2

Wri

te a

mat

rix

equ

atio

n f

or e

ach

sys

tem

of

equ

atio

ns.

1.�

3x�

2y�

92.

6x�

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10

� �

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7

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5.3x

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5x

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3m�

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17

� �

�

�

Sol

ve e

ach

mat

rix

equ

atio

n o

r sy

stem

of

equ

atio

ns

by

usi

ng

inve

rse

mat

rice

s.

7.�

�(2

,�4)

8.�

�(5

,1)

9.�

�(3

,�5)

10.

��

(1,7

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Pre-

Act

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ow a

re i

nve

rse

mat

rice

s u

sed

in

pop

ula

tion

eco

logy

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d th

e in

trod

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son

4-8

at

the

top

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age

202

in y

our

text

book

.

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2 �

2 m

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atio

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iven

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on

1.a.

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rix

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ns.

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lain

how

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use

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e m

atri

x eq

uat

ion

you

wro

te a

bove

to

solv

e th

e sy

stem

.Use

as

few

mat

hem

atic

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ymbo

ls i

n y

our

expl

anat

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as

you

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ual

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olve

th

esy

stem

.

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ple

an

swer

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ind

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e th

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th

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con

d r

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will

be

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valu

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at c

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mem

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orde

r in

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to

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e th

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mat

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at m

ake

up

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emem

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VC

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r “c

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x is

on

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ide

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equ

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n,j

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as t

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4-8

4-8


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12. A

D

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Chapter 4 Assessment Answer Key Form 1 Form 2APage 217 Page 218 Page 219

(continued on the next page)


13.

14.

15.

16.

17.

18.

19.

20.

B:

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

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20.

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A

B

D

Chapter 4 Assessment Answer Key Form 2A (continued) Form 2BPage 220 Page 221 Page 222

An

swer

s


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

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impossible

(3, �4)

Chapter 4 Assessment Answer Key Form 2CPage 223 Page 224

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Chapter 4 Assessment Answer Key Form 2DPage 225 Page 226

An

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10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

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(�2, 5)

Chapter 4 Assessment Answer Key Form 3Page 227 Page 228

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Chapter 4 Assessment Answer KeyPage 229, Open-Ended Assessment

Scoring Rubric


Score General Description Specific Criteria

• Shows thorough understanding of the concepts of adding,subtracting, and multiplying matrices; multiplying by ascalar; transforming geometric figures using matrices;evaluating determinants; solving systems of equationsusing Cramer’s Rule; identifying and finding inversematrices; and using inverse matrices to solve matrixequations.

• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Goes beyond requirements of some or all problems.

• Shows an understanding of the concepts of adding,subtracting, and multiplying matrices; multiplying by a scalar;transforming geometric figures using matrices; evaluatingdeterminants; solving systems of equations using Cramer’sRule; identifying and finding inverse matrices; and usinginverse matrices to solve matrix equations.

• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Satisfies all requirements of problems.

• Shows an understanding of most of the concepts ofadding, subtracting, and multiplying matrices; multiplyingby a scalar; transforming geometric figures using matrices;evaluating determinants; solving systems of equationsusing Cramer’s Rule; identifying and finding inversematrices; and using inverse matrices to solve matrixequations.

• May not use appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are satisfactory.• Satisfies the requirements of most of the problems.

• Final computation is correct.• No written explanations or work is shown to substantiate

the final computation.• Satisfies minimal requirements of some of the problems.

• Shows little or no understanding of most of the concepts ofadding, subtracting, and multiplying matrices; multiplyingby a scalar; transforming geometric figures using matrices;evaluating determinants; solving systems of equationsusing Cramer’s Rule; identifying and finding inversematrices; and using inverse matrices to solve matrixequations.

• Does not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are unsatisfactory.• Does not satisfy requirements of problems.• No answer may be given.

0 UnsatisfactoryAn incorrect solutionindicating no mathematicalunderstanding of theconcept or task, or nosolution is given

1 Nearly Unsatisfactory A correct solution with nosupporting evidence orexplanation

2 Nearly SatisfactoryA partially correctinterpretation and/orsolution to the problem

3 SatisfactoryA generally correct solution,but may contain minor flawsin reasoning or computation

4 SuperiorA correct solution that is supported by well-developed, accurateexplanations

An

swer

s


Chapter 4 Assessment Answer Key Page 229, Open-Ended Assessment

Sample Answers

1. Student responses should include:1a. A and B must have the same

dimensions, so m � j and n � k.1b. Inner dimensions must be equal. Thus,

n � j for AB and k � m for BA.1c. The determinant of A exists only if A is

a square matrix, so m � n.1d. Matrix size for scalar multiplication is

irrelevant, so there are no restrictionson j and k.

1e. Only square matrices (potentially) haveinverses, so m � n if A has an inverse.

2a. Student matrices must be of the form

V � � �, where P1(x1, y1),

P2(x2, y2), and P3(x3, y3) are the points,no two in the same quadrant, which thestudent has chosen. All students shouldstate that V is a 2 � 3 matrix.

2b. Student matrices must be of the formT � V, which represents a translation ofthe original triangle 3 units right and 4units down.

T � V � � �2c. Student matrices must be of the form

3V, which represents a dilation of theoriginal triangle which triples thelength of each side.

3V � � �2d. Student matrices must be of the form

RV, which represents a reflection of theoriginal triangle over the y-axis.

RV � � �

2e. Student matrices must be of the formCV, which represents a 90�counterclockwise rotation of the originaltriangle about the origin.

CV � � �2f. Students should indicate that I is the

identity matrix, so that IV � V. Thismeans that the image and the preimageof the triangle are exactly the same inall respects.

3a. Sample answer: Expansion using thebottom (3rd) row would be easiest sincethe elements are 1, 0 and �1.

3b. Student responses should indicate thereason for choosing one method over theother.

3c. Students should correctly apply thechosen method to arrive at the solutiona � �5.

4a. 3c � 2v � 85 2c � v � 50(Variables may vary); c: the cost of oneCD, v: the cost of one video

4b. Students should correctly applyCramer’s Rule to determine that c � 15and v � 20, meaning that the cost of oneCD is $15 and the cost of one video is$20.

4c. � � � � � � � �; Students should

demonstrate the correct method ofsolving the equation using inversematrices, and again conclude that c � 15and v � 20.

4d. Student responses should indicate thereason for choosing one method over theother.

8550

cv

3 22 1

�y1 �y2 �y3x1 x2 x3

�x1 �x2 �x3y1 y2 y3

3x1 3x2 3x33y1 3y2 3y3

x1 � 3 x2 � 3 x3 � 3y1 � 4 y2 � 4 y3 � 4

x1 x2 x3y1 y2 y3

In addition to the scoring rubric found on page A31, the following sample answers may be used as guidance in evaluating open-ended assessment items.


1. Cramer’s Rule

2. square matrix

3. transformation

4. matrix

5. reflection

6. Expansion byminors

7. minor

8. isometry

9. vertex matrix

10. dilation

11. Sample answer:An identity matrix is a square matrixin which theelements on themain diagonal areall ones and all theother elements arezeroes.

12. Sample answer:A column matrix isa matrix that hasexactly one column.

1.

2.

3.

4.

5.

Quiz (Lessons 4–3 and 4–4)

Page 231

1.

2.

3.

4.

5.

1.

2.

3.

4.

5.

Quiz (Lessons 4–7 and 4–8)

Page 232

1.

2.

3.

4.

5. (4, 5)

no inverse exists

yes

(�2, 3, 1)

(�1, 7)

24 units2

52

7

X�(3, 2), Y�(2, �3),Z�(�5, 4)

A(�2, �4), D�(1, 6)

3 � 4

(3, �1)

Chapter 4 Assessment Answer Key Vocabulary Test/Review Quiz (Lessons 4–1 and 4–2) Quiz (Lessons 4–5 and 4–6)

Page 230 Page 231 Page 232

An

swer

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xyz

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1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

1

impossible

Sample answer:a � b � c � 37

b � a � 4b � c � 29

yes

(1, 2)

y

xO

D � all real numbers;{R � y � y � �4}

�45

�

5.9

5050

impossible

[�17 �48 �27 30]

D

B

A

A

C

Chapter 4 Assessment Answer Key Mid-Chapter Test Cumulative ReviewPage 233 Page 234

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2.

3.

4.

5.

6.

7.

8.

9.

10.

11. 12.

13. 14.

15.

16.

17.

18. DCBA

DCBA

DCBA

DCBA

0 0 0

.. ./ /

.

99 9 987654321

87654321

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5 4

0 0 0

.. ./ /

.

99 9 987654321

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1 5

0 0 0

.. ./ /

.

99 9 987654321

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1 / 6

0 0 0

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Chapter 1 Assessment Answer KeyStandardized Test Practice

Page 235 Page 236

An

swer

s


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25. (3, �4)

(2, �3)

�19

D�(5, 3); E�(4, �8);F�(�2, 1)

16 bedroom sets,5 living room sets

y

xO

4

8

12

4 8 12 16

(6, 10)

(16, 5)

(6, 5)

b � 6; � � 521b � 42� 546

(2, 1)

(�1, �3)

inconsistent

�3

Sample answer using (2, 16) and (3, 10):n � �6p � 28; 1

n

p

68

0

1012141618

1 2 3 4Price (dollars)

Num

ber

Sol

d

y � �2x � 7

y � �45

�x � �153�

19

�4�5 �3 �2 �1 0 2 3 4 5 61

all real numbers

0 1�3 �2 �1�4

�x � x �3 or x ��12

��

g � the number of additional games to

be won; �g

2�0

9� � 0.75;

at least 6 games

3

28

N, W, Z, Q, R

�19

Chapter 4 Assessment Answer Key Unit 1 TestPage 237 Page 238

� �7 �4 1015 1 �9

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Documents

Chapter 4 Resource Masters - KTL MATH CLASSES©Glencoe/McGraw-Hill iv Glencoe Algebra 2 Teacher’s Guide to Using the Chapter 4 Resource Masters The Fast FileChapter Resource system