4.7 Inverse Matrices and Systems. 1) Inverse Matrices and Systems of Equations You have solved systems of equations using graphing, substitution, elimination…oh.

4.7 Inverse Matrices and Systems

1) Inverse Matrices and Systems of Equations

You have solved systems of equations using graphing, substitution, elimination…oh my…

In the “real world”, these methods take too long and are almost never used.

Inverse matrices are more practical.

1) Inverse Matrices and Systems of Equations

For a System of Equations

1453

52

yx

yx

1) Inverse Matrices and Systems of Equations

For a We can write a System of Equations Matrix Equation

1453

52

yx

yx

14

5

53

21

y

x

1) Inverse Matrices and Systems of Equations

Example 1:Write the system as a matrix equation

62

1132

yx

yx

1) Inverse Matrices and Systems of Equations

Example 1:Write the system as a matrix equation

Matrix Equation

62

1132

yx

yx

6

11

21

32

y

x

1) Inverse Matrices and Systems of Equations

Example 1:Write the system as a matrix equation

Matrix Equation

62

1132

yx

yx

6

11

21

32

y

x

Coefficient matrix

Constant matrix

Variable matrix

1) Inverse Matrices and Systems of Equations

Example 2:

822

52

0

zyx

zyx

zyx

1) Inverse Matrices and Systems of Equations

Example 2:

8

5

0

212

121

111

z

y

x

822

52

0

zyx

zyx

zyx

1) Inverse Matrices and Systems of Equations

Example 2:

A BX

8

5

0

212

121

111

z

y

x

822

52

0

zyx

zyx

zyx

1) Inverse Matrices and Systems of Equations

BAX

1) Inverse Matrices and Systems of Equations

BAX 1

When rearranging, take the inverse of A

BAX

1) Inverse Matrices and Systems of Equations

The Plan…

“Solve the system” using matrices and inverses

BAX 1BAX

1) Inverse Matrices and Systems of Equations

Example 3:Solve the system

62

1132

yx

yx

1) Inverse Matrices and Systems of Equations

Example 3:Solve the system

Step 1: Write a matrix equation

62

1132

yx

yx

1) Inverse Matrices and Systems of Equations

Example 3:Solve the system

Step 1: Write a matrix equation

6

11

21

32

y

x

62

1132

yx

yx

1) Inverse Matrices and Systems of Equations

Example 3:Solve the system

Step 2: Find the determinant and A-1

62

1132

yx

yx

1) Inverse Matrices and Systems of Equations

Example 3:Solve the system

Step 2: Find the determinant and A-1

21

32A

62

1132

yx

yx

Change signs

Change places

1) Inverse Matrices and Systems of Equations

Example 3:Solve the system

Step 2: Find the determinant and A-1

21

32A

62

1132

yx

yx

Change signs

Change places

detA = 4 – 3

= 1

1) Inverse Matrices and Systems of Equations

Example 3:Solve the system

Step 2: Find the determinant and A-1

21

32

21

32

1

1

21

32

1

1

A

A

A

62

1132

yx

yx

1) Inverse Matrices and Systems of Equations

Example 3:Solve the system

Step 3: Solve for the variable matrix

62

1132

yx

yx

1) Inverse Matrices and Systems of Equations

Example 3:Solve the system

Step 3: Solve for the variable matrix

BAy

x

BAX

1

1

62

1132

yx

yx

1) Inverse Matrices and Systems of Equations

Example 3:Solve the system

Step 3: Solve for the variable matrix

1

4

6

11

21

32

1

1

y

x

y

x

BAy

x

BAX

62

1132

yx

yx

1) Inverse Matrices and Systems of Equations

Example 3:Solve the system

Step 3: Solve for the variable matrix

1

4

6

11

21

32

1

1

y

x

y

x

BAy

x

BAX

The solution to the system is (4, 1).

62

1132

yx

yx

1) Inverse Matrices and Systems of Equations

Example 4:Solve the system. Check your answer.

523

735

ba

ba

1) Inverse Matrices and Systems of Equations

Example 4:Solve the system. Check your answer.

5

7

23

35

b

a

523

735

ba

ba

1) Inverse Matrices and Systems of Equations

Example 4:Solve the system. Check your answer.

523

735

ba

ba

53

32

53

32

1

1

23

35

1

1

A

A

A

detA = 10 - 9

= 1

1) Inverse Matrices and Systems of Equations

Example 4:Solve the system. Check your answer.

523

735

ba

ba

4

1

5

7

53

32

1

1

b

a

b

a

BAb

a

BAX

1) Inverse Matrices and Systems of Equations

Example 4:Solve the system. Check your answer.

523

735

ba

ba

The solution to the system is (-1, 4).

4

1

5

7

53

32

1

1

b

a

b

a

BAb

a

BAX

1) Inverse Matrices and Systems of Equations

Example 4:Solve the system. Check your answer.

Check

523

735

ba

ba

77

7125

7)4(3)1(5

735

ba

55

583

5)4(2)1(3

523

ba

What about a matrix that has no inverse?

It will have no unique solution.

1) Inverse Matrices and Systems of Equations

1) Inverse Matrices and Systems of Equations

Example 5:Determine whether the system has a unique solution.

842

52

yx

yx

1) Inverse Matrices and Systems of Equations

Example 5:Determine whether the system has a unique solution.

Find the determinant.

842

52

yx

yx

1) Inverse Matrices and Systems of Equations

Example 5:Determine whether the system has a unique solution.

Find the determinant.

842

52

yx

yx

8

5

42

21

y

x

1) Inverse Matrices and Systems of Equations

Example 5:Determine whether the system has a unique solution.

Find the determinant.

0

)2(2)4(1

42

21det

42

21

A

A

842

52

yx

yx

Since detA = 0, there is no inverse.

The system does not have a unique solution.

Homework

p.217 #1-5, 7-10, 20, 21, 23, 24, 26, 27, 36

DUE TOMORROW: Two codes

TEST: Wednesday Nov 25Chapter 4