Section 4Chapter 4. 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objectives 2 5 3 4 Solving Systems of Linear Equations by Matrix Methods Define

Section 4Chapter 4

1

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Objectives

2

5

3

4

Solving Systems of Linear Equations by Matrix Methods

Define a matrix.

Write the augmented matrix of a system.

Use row operations to solve a system with two equations.

Use row operations to solve a system with three equations.

Use row operations to solve special systems.

4.4


Define a matrix.

Objective 1

Slide 4.4- 3


A matrix is an ordered array of numbers.

The numbers are called elements of the matrix.

Matrices are named according to the number of rows and columns they contain.

The number of rows followed by the number of columns give the dimensions of the matrix.

2 3 5

7 1 2

Rows

Columns

Slide 4.4- 4

Define a matrix.


A square matrix is a matrix that has the same number of rows as columns.

1 0

1 2

8 1 3

2 1 6

0 5 3

5 9 7

2 2 matrix

4 3 matrix

Slide 4.4- 5

Define a matrix.



Objective 2

Slide 4.4- 6


An augmented matrix has a vertical bar that separates the columns of the matrix into two groups.

3 1

2 5

x y

x y

1 3 1

2 1 5

Slide 4.4- 7



Matrix Row Operations

1. Any two rows of the matrix may be interchanged.

2. The elements of any row may be multiplied by any nonzero real number.

3. Any row may be changed by adding to the elements of the row the product of a real number and the corresponding elements of another row.

Slide 4.4- 8



Examples of Row Operations

Row operation 1

Row operation 2

1 0 7

1 0 7

4 8 3 4

9

8

2

3

2 3 9

3

Interchange row 1 and row 3.

2 3 9 6 9 27

4 8 3 4 8 3

1 0 7 1 0 7

Multiply row 1 by 3.

Slide 4.4- 9



Examples of Row Operations (continued)

Row operation 3

2 3 9 0

4 8 3 4 8 3

1 0 7 1 0

5

7

3

Multiply row 3 by –2; add them to the corresponding numbers in row 1.

Slide 4.4- 10




Objective 3

Slide 4.4- 11


Row operations can be used to rewrite a matrix until it is the matrix of a system whose solution is easy to find. The goal is a matrix in the form

for systems with two and three equations.

A matrix written as shown above with a diagonal of ones, is said to be in row echelon form.

1

0 1

a b

c

1

0 1

0 0 1

a b c

d e

f

Slide 4.4- 12



Use row operations to solve the system.

Write the augmented matrix of the system.

Use row operations to change the matrix into one that leads to a system that is easy to solve.

It is best to work by columns.

2 9

3 13

x y

x y

1 2 9

3 1 13

Slide 4.4- 13

CLASSROOM EXAMPLE 1

Using Row Operations to Solve a System with Two Variables

Solution:


–3R1 + R2 2 9

3 13

x y

x y

1 2 9

3 (1)( 3) 1 ( 2)( 3) 13 ( 3)(9)

1 2 9

3 1 13

127

2 9

7 10 4

1

R

Original number from row 2

3 times the number from row 1

2 9

2

1

0 1

Slide 4.4- 14

Using Row Operations to Solve a System with Two Variables (cont’d)CLASSROOM EXAMPLE 1


The matrix gives the system

Substitute –2 for y in the first equation.

2 9

3 13

x y

x y

2 9

2

x y

y

2 9

2

x y

y

2( ) 92x

4 9x 5x

The solution set is {(5, –2)}.

Slide 4.4- 15

Using Row Operations to Solve a System with Two Variables (cont’d)CLASSROOM EXAMPLE 1


Use row operations to solve a system with three equations.

Objective 4

Slide 4.4- 16



Interchange rows 1 and 2.


2 7

3 7

5 9

x y z

x y z

x y z

1 3 1 7

2 1 1 7

1 1 5 9

3 7

2 7

5 9

x y z

x y z

x y z

Slide 4.4- 17

Using Row Operations to Solve a System with Three VariablesCLASSROOM EXAMPLE 2

Solution:



1 2

1 3 1 7

2 1 1 7

1 1 5 9

2R R

1 3 1 7

0 5 3 7

1 1 5 9

1 3

1 3 1 7

0 5 3 7

1 1 5 9 R R

1 3 1 7

0 5 3 7

0 2 6 2

Slide 4.4- 18

Using Row Operations to Solve a System with Three Variables (cont’d)CLASSROOM EXAMPLE 2


125

1 3 1 7

0 5 3 7

0 2 6 2

R

3 75 5

1 3 1 7

0 1

0 2 6 2

3

3

75 5

2

1 3 1 7

0 1

0 2 6 2 2R R

3 75 5

24 245 5

1 3 1 7

0 1

0 0

3 75 5

524 245 3245

1 3 1 7

0 1

0 0 R

3 75 5

1 3 1 7

0 1

0 0 1 1

Slide 4.4- 19



3 75 5

1 3 1 7

0 1

0 0 1 1

This matrix gives the system

3 7

3 7

5 51.

x y z

y z

z

Substitute 1 for z in the second equation.

13 7

( )5 5

y

2y

Slide 4.4- 20



The solution set is {(2, – 2, 1)}.

Substitute 2 for y and1 for z in the first equation.

3 7

3( 2) 1 7

x y z

x

5 7x 2x

Slide 4.4- 21



Use row operations to solve special systems.

Objective 5

Slide 4.4- 22



Write the augmented matrix.

2

2 2 2

x y

x y

1 1 2

2 2 2

1 2

1 1 2

2 2 2 2R R

1 1 2

0 0 6

Slide 4.4- 23

Recognizing Inconsistent Systems or Dependent EquationsCLASSROOM EXAMPLE 3

Solution:



The false statement indicates that the system is inconsistent and has no solution.

The solution set is .

2

0 6.

x y

Slide 4.4- 24

Recognizing Inconsistent Systems or Dependent Equations (cont’d)CLASSROOM EXAMPLE 3



Write the augmented matrix.

2

2 2 4

x y

x y

1 1 2

2 2 4

1 2

1 1 2

2 2 24 R R

1 1 2

0 0 0

Slide 4.4- 25


Solution:



The true statement indicates that the system has dependent equations.

The solution set is {(x, y)| x – y = 2}.

2

0 0.

x y

Slide 4.4- 26


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Section 4Chapter 4. 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objectives 2 5 3 4 Solving Systems of Linear Equations by Matrix Methods Define