Section 13-5: Inverses of Matrices

Objectives:

1) Background – what are inverses and why find them.

2) Process for finding the inverse of a 2x2 matrix.

Background: Inverses of matrices are useful for solving systems of equations.

To introduce the concept, consider the following system of equations; this system can be written as a matrix.

The solution to this system can also be written as a matrix.

The process for using an inverse to solve a system of equation is similar to the process for solving the following simple equation.

2

3 4

6

4

x y

x y

1

4

2

43

6x

y

4

2

x

y

1 0 4

0 1 2

x

y

5 10x 1

5 5

1

5x 10 1 2x

The result of multiplying 5 by its reciprocal was 1.

The result of multiplying a matrix by its inverse

(similar to a reciprocal) is the identity matrix.

The question becomes, how do we find the inverse matrix??

Note: If a matrix is defined as A, then its inverse is defined as A-1.

1

55

1

5x 10 1 2x

Background (Continued)1

5

Inverse 2 1 Inverse 6

Matrix 3 4 Matrix 2

1 0

0

4

4 1

x x

y y

IdentityMatrix

I

1 1A A A A I

2

3 4

6

4

x y

x y

Your book (see pg. 590) provides a formula for finding the inverse of a 2x2 matrix. The formula is:

Example:

Finding the Inverse Matrix of a 2x2

1 1If and 0, then

a b d bA A A

c d c aA

2 1

4 0A

First step, compute A 4A

1Finally, multiply the scalar by modified .A

A

0 1 11

1 14 41

4 1

41 1

4 2 11

14 4 1 2

0 1 0

4 2 1A

Is the product of a matrix and its inverse the identity matrix?

Silly Question: What is the identity matrix for a 3x3 matrix?

Check

2 1

4 0A

141

12

0

1A

2 1

4 0A

0 1 1 14 2

2 0 1 1 1

4 0 0 1 0 1 1

4 22 1 0 1 1

4 24 0 1

1 1 0

0 1A A

1 1 0Is ?

0 1A A I