Table of Contents

Matrices - Inverse of a 3 3 Matrix

• Recall that to find an inverse of a 2 2 matrix, apply the formula ...

ac

bd

bcad

1A 1

• The formula for the inverse of a 3 3 matrix is not so simple. Another method is as follows.

Table of Contents Slide 2

Matrices - Inverse of a 3 3 Matrix

(1) Write the augmented matrix, with matrix A on the left, and the identity matrix I on the right ...

IA

(2) Use elementary row operations on the matrix, until the identity matrix is on the left ...

I

1AI (3) The resulting matrix on the right is A-1 ...

Table of Contents Slide 3

Matrices - Inverse of a 3 3 Matrix

• Example: Let matrix A be given on the right. Find A-1.

032

412

321

A

• Form the augmented matrix ...

100

010

001

032

412

321

Table of Contents Slide 4

Matrices - Inverse of a 3 3 Matrix

25.005.04.0100

5.03.04.0010

25.045.06.0001

• Using elementary row operations to get the identity on the left yields ...

• A-1 is the resulting matrix on the right.

25.005.04.0

5.03.04.0

25.045.06.0

A

Table of Contents Slide 5

Matrices - Inverse of a 3 3 Matrix

• In the example the elementary row operations were not shown. It takes considerable effort and attention to detail. Since the process is so long and tedious, finding inverses on the calculator is the much preferred method.

Table of Contents

Matrices - Inverse of a 3 3 Matrix