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

145352

yxyx

1) Inverse Matrices and Systems of Equations

For a We can write a System of Equations Matrix Equation

145352

yxyx

145

5321yx

1) Inverse Matrices and Systems of Equations

Example 1:Write the system as a matrix equation

621132

yxyx

1) Inverse Matrices and Systems of Equations

Example 1:Write the system as a matrix equation

Matrix Equation

621132

yxyx

611

2132yx

1) Inverse Matrices and Systems of Equations

Example 1:Write the system as a matrix equation

Matrix Equation

621132

yxyx

611

2132yx

Coefficient matrix

Constant matrix

Variable matrix

1) Inverse Matrices and Systems of Equations

Example 2:

822520

zyxzyxzyx

1) Inverse Matrices and Systems of Equations

Example 2:

850

212121111

zyx

822520

zyxzyxzyx

1) Inverse Matrices and Systems of Equations

Example 2:

A BX

850

212121111

zyx

822520

zyxzyxzyx

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

621132

yxyx

1) Inverse Matrices and Systems of Equations

Example 3:Solve the system

Step 1: Write a matrix equation

621132

yxyx

1) Inverse Matrices and Systems of Equations

Example 3:Solve the system

Step 1: Write a matrix equation

611

2132yx

621132

yxyx

1) Inverse Matrices and Systems of Equations

Example 3:Solve the system

Step 2: Find the determinant and A-1

621132

yxyx

1) Inverse Matrices and Systems of Equations

Example 3:Solve the system

Step 2: Find the determinant and A-1

2132

A

621132

yxyx

Change signs

Change places

1) Inverse Matrices and Systems of Equations

Example 3:Solve the system

Step 2: Find the determinant and A-1

2132

A

621132

yxyx

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

2132

2132

11

2132

1

1

A

A

A

621132

yxyx

1) Inverse Matrices and Systems of Equations

Example 3:Solve the system

Step 3: Solve for the variable matrix

621132

yxyx

1) Inverse Matrices and Systems of Equations

Example 3:Solve the system

Step 3: Solve for the variable matrix

BAyx

BAX

1

1

621132

yxyx

1) Inverse Matrices and Systems of Equations

Example 3:Solve the system

Step 3: Solve for the variable matrix

14

611

2132

1

1

yx

yx

BAyx

BAX

621132

yxyx

1) Inverse Matrices and Systems of Equations

Example 3:Solve the system

Step 3: Solve for the variable matrix

14

611

2132

1

1

yx

yx

BAyx

BAX

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

621132

yxyx

1) Inverse Matrices and Systems of Equations

Example 4:Solve the system. Check your answer.

523735

baba

1) Inverse Matrices and Systems of Equations

Example 4:Solve the system. Check your answer.

57

2335ba

523735

baba

1) Inverse Matrices and Systems of Equations

Example 4:Solve the system. Check your answer.

523735

baba

5332

5332

11

2335

1

1

A

A

A

detA = 10 - 9

= 1

1) Inverse Matrices and Systems of Equations

Example 4:Solve the system. Check your answer.

523735

baba

41

57

5332

1

1

ba

ba

BAba

BAX

1) Inverse Matrices and Systems of Equations

Example 4:Solve the system. Check your answer.

523735

baba

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

41

57

5332

1

1

ba

ba

BAba

BAX

1) Inverse Matrices and Systems of Equations

Example 4:Solve the system. Check your answer.

Check

523735

baba

7771257)4(3)1(5735

ba

555835)4(2)1(3523

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.

84252

yxyx

1) Inverse Matrices and Systems of Equations

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

Find the determinant.

84252

yxyx

1) Inverse Matrices and Systems of Equations

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

Find the determinant.

84252

yxyx

85

4221yx

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

4221

det

4221

A

A

84252

yxyx

Since detA = 0, there is no inverse.

The system does not have a unique solution.

Homeworkp.217 #1-5, 7-10, 20, 21, 23, 24, 26,

27, 36

DUE TOMORROW: Two codes

TEST: Wednesday Nov 25Chapter 4