A: Matrices, Basic Matrix Operations€¦ · A: Matrices, Basic Matrix Operations (Lessons 2.3 & 2.4 pg 86 – 107) A matrix ¬¼is a rectangular array of numbers like: This matrix

Finite Math B Chapter 2 + Supplements: MATRICES 1

Chapter 2 and Supplements MATRICES

A: Matrices, Basic Matrix Operations (Lessons 2.3 & 2.4 pg 86 – 107)

A matrix is a rectangular array of numbers like: This matrix has _____ rows and _____ columns.

When we want to describe a particular entry in an m-by-n matrix,

we use the notation ,i ja where ( , ) ( , )i j row column position.

For example, in the matrix above:

1,2a = 2,3a =

The number -1 is in the _________ position.

A matrix with an equal number of rows and columns is called a “Square Matrix” A matrix with only one row is called a “Row Matrix.” A matrix with only one column is called a “Column Matrix.” Example 1: State the size of each matrix. State if the matrix is a square, row, or column matrix when applicable.

a) 1 3 4 8 b)

2 4 6

3 5 9

8 1 0

c)

9 3

2 8

1 8

d)

2

1

0

Adding/Subtracting Matrices If two matrices are the same size (mxn) then you can add or subtract them by combining corresponding elements.

3 5 1

2 4 7

http://en.wikipedia.org/wiki/File:Matrix.svg


Example 2: Add or subtract the following matrices. Write “undefined” for expressions that are undefined.

a. 2 4 3 0

5 9 4 8

b.

3 4 2 9

5 6 4 8

7 8 2 3

9 10 1 11

c.

3 4 15 1

0 7 81 7

3 6 4

d. 1 2 0 8 0 7 5 9

The “Additive Inverse” of a matrix X is a matrix –X such that X + -X = 0 Example 3: What is the Additive inverse of the following matrices?

a) 1 3 4 8 b)

9 3

2 8

1 8

Setting up a matrix to represent a “real life” problem. Example 4:

A trendy garment company receives orders from three

clothing shops. The first shop orders 25 jackets, 75

shirts, and 75 pairs of pants. The second shop orders 30

jackets, 50 shirts, and 50 pairs of pants.. The third shop

orders 20 jackets, 40 shirts, and 35 pairs of pants.

Display this information as a matrix where the rows

represent shops and the columns represent types of

garments.


Example 5:

Example 6:

You are also told that at Gina’s an extra pizza topping is

$0.50, but at Toni’s it is only $0.30. In addition, at

Gina’s salad dressing is included, but at Toni’s salad

dressing is $1.00 extra. Write a new 3x2 matrix that

shows the costs of a pizza with an additional topping and

a salad with dressing.

The National League batting leaders for 2003 had the following batting statistics:

The following show the statistics for the same three players at the end of the 2004 season.

Find and label a matrix that displays the CHANGES in these statistics from 2003 to 2004. Notice that several of the statistics decreased. How will you show that in your matrix?

You are planning dinner and decide to call several pizza

shops and get prices. Gina’s Pizzeria charges $12.16 for

a large one-topping pizza, $1.15 for a 2-liter of soda, and

$4.05 for a family sized salad. Toni’s Pizza charges

$10.86 for a pizza, $0.89 for soda, and $3.89 for salad.

Display this information as a 3 x 2 matrix.


B: Scalar Factors, Determinants, Graphing Calculators (Lessons 2.4 p96 and supplements)

Multiplication by a Scalar Factor: Multiply each entry in the matrix by the factor

b c xb xc

xd e xd xe

Example 1: Simplify.

a.

0

3 8

5

b.

1 7 3

2 4 5 3

1 0 8

Example 2:

Example 3:


Determinants A determinant is a special number associated with a square matrix. The determinant has several useful applications in algebra and other advanced mathematics courses. Notation:

Matrix Notation (square brackets) 2x2 a b

c d

3x3

a b c

d e f

g h i

Determinant Notation (straight brackets) 2x2 a b

c d 3x3

a b c

d e f

g h i

(This notation represents “find the determinant of the matrix”) Also: det(A) or detA represents “the determinant of matrix A” 2x2 Matrices The determinant of a 2 x 2 matrix is found as follows: Example 4: Find the determinant of each 2 x 2 matrix

a) 3 4

5 6

b) 1 3

6 7

Example 5: Evaluate.

a) 2 3

5 10

b)

2 7

3 5


3x3 Matrices The determinant of a 3x3 matrix is found as follows: This is easier to find using a little trick called “The Rule of Sarrus”: 1. Copy the first two columns next to column 3. 2. Multiply the three “southeast” (downward)diagonals. Find the sum. 3. Multiply the three “northeast” (upward) diagonals. Find the sum. 4. Determinant = (SE sum) – (NE sum) bottom – top Example 6: Find the determinant of each matrix.

a)

1 3 5

2 1 3

4 5 2

b)

2 3 1

4 2 3

2 2 1

Example 7: Evaluate

1 2 3

1 2 3

3 2 1


Using Technology

Working with Matrices in the TI 83/TI83plus/TI84 Graphing Calculator **NOTE: If you have your OWN graphing calculator, you are permitted to use it on exams. Otherwise, you will NOT BE PROVIDED with a graphing calculator on exam day. There are no items on the exam that will require the use of this technology**

TI-83 Matrix Menu

Note: Some TI 83 models do not require you to “shift” to access the matrix menu.

BASIC OPERATIONS Use the graphing calculator to perform the following operations:

1. 2 4 3 0

5 9 4 8

2.

2 4 3 0

5 9 4 8

3. 2 4 1

5 9 4

4. 2 4

55 9

MATRX 2nd x-1

About the Matrix (MATRX) menu:

NAMES – used to paste the name of a matrix into the home screen or into a program.

MATH – contains all the operations that can be done with a matrix.

EDIT – where you set the size of the matrix and enter the elements.


DETERMINANTS In the matrix menu, use the EDIT screen to enter your matrix. Use 2nd (MODE) to quit to the main screen. In the matrix menu, use the MATH screen to paste det( into your home screen and the NAMES screen to paste the name of the matrix. Close ) and hit enter

Use the graphing calculator to find the determinant of each of the following matrices.

5.

2 3

5 10

6.

1 3 5

2 1 3

4 5 2

C: Multiplying Matrices (Lesson 2.4 pg 97 – 107) Let A be an m x n matrix Let B be an n x k matrix The product matrix AB is an m x k matrix. Example 1: The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, whenever these products exist. a) A is 4 x 4, B is 4 x 2 b) A is 3 x 1, B is 3 x 1 c) A is 2 x 3, B is 1 x 2


Multiplying Matrices Let A be an m x n matrix Let B be an n x k matrix To find the product: Multiply each element in the row of A by the corresponding element in the column of B, then add these products. The product matrix AB is an m x k matrix.

SAMPLE 2: Find the product

SAMPLE 1: Find the product AB

2 3 1

4 2 2A

1

8

6

B

A is 2 x 3 B is 3 x 1 -------> AB will be 2 x 1 Step 1: Multiply the elements of the first row of A and the corresponding elements in the column of B.

2(1) + 3(8) + -1(6) = 20

Step 2: Multiply the elements of the second row of A and the corresponding elements in the column of B.

4(1) + 2(8) + 2(6) = 32

Step 3: Write your solution as a 2 x 1 matrix. 20

32


Example 2: Find the product. Write “undefined for expressions that are undefined.

a)

22 4 6

31 3 0

4

b) 2 1 3

5 8 2

c) 2 1 7 3

3 6 1 4

d)

0 22 2 1

1 43 0 1

0 2


e)

1 1 2 2 3

3 2 1 3 2

4 5 2 4 5

f) 2 5 3 1 2 3

7 4 6 3 2 1

g)

3

2 2 4 6 1

1


D: Geometric Transformations using Matrices

We can use matrices to represent geometric figures:

A transformation is a change made to a figure. The original figure is called the preimage, while the transformed figure is called the image.

TRANSLATION When we slide a figure without changing the size or shape of the figure, it is said to be a translation. By using matrix addition, we can translate the vertices of a figure. Example 1: Given a triangle ABC where A(-4,1), B (-2,5), and C(0,2), translate the preimage 5 units right and 3 units down. Then sketch the image Preimage Matrix + Translation Matrix = new image Example 2: Given a quadrilateral ABC where A(3,0), B(5,2), C(5,4), and D(2,4). Translate the preimage six units left and 2 units up.

x

y


DILATION A dilation is a transformation that changes the size of the preimage. Dilation factor x preimage matrix = new image Example3: Given a triangle ABC where A(-2,0), B(0,4) and C(2,1), increase the triangle by a factor of 2. Then sketch the image. Example 4: Given the quadrilateral with vertices A(4, 3), B(0,5), C(-10,14) and D(1, 1.5), increase the figure by 150%. State the vertices of the new image. REFLECTIONS (FLIPS) A reflection, or flip, is a transformation that creates symmetry on the coordinate plane. A reflection maps a point or figure in the coordinate plane to its mirror image using a specific line as its line of reflection. Usually we use the following lines of reflection:

Reflection Matrix x Preimage Matrix = New Image Example 5: Given quadrilateral A(2,1), B(8,1), C(8,4), and D(5,5), find the coordinates of the image after a reflection: a) Across the y-axis b) Across y = x

x

y


ROTATIONS (SPINS) A rotation turns a figure around a fixed point. We usually consider the rotation to be clockwise around the origin as the fixed point.

Rotation Matrix x Preimage Matrix = New Image Example 6: Rotate the triangle with vertices A(1,1), B (5,2), and C(-2,3) by the stated amount. Give the vertices of the new image. a) 90 degrees b) 180 degrees Example 7: A triangle has vertices A(1,1), B(5,2), and C(-2,3). First, The triangle is rotated 90 degrees. Next, the triangle is reflected across the y-axis. Finally, the triangle is translated right 3 units and down 4 units. What are the FINAL coordinates of the image?

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A: Matrices, Basic Matrix Operations€¦ · A: Matrices, Basic Matrix Operations (Lessons 2.3 & 2.4 pg 86 – 107) A matrix ¬¼is a rectangular array of numbers like: This matrix