4.4: Identity and Inverse Matrices

Objectives:Students will be able to find and use an inverse matrix.

Identity Matrix

1 is the multiplicative identify for real #’s : 1· a=a and a· 1 = a

For matrices n X n, the identity matrix has 1’s on its main diagonals and 0’s elsewhere

2 X 2 IDENTITY MATRIX 3 X 3 IDENTITY MATRIX

10

01

100

010

001

If A is an n x n matrix and I is the identity matrix, then AI = A and IA = A

2 matrices are Inverse Matrices if the product of the two n X n matrices in both orders is the n X n identity.

Example: Prove that the two matrices are inverses.

61

173:;

31

176: BA

AB: BA:

To find the Inverse of a 2 x 2 matrix:

)0(

111

cdad

ac

bd

cbadac

bd

AA

dc

baA

Example: Find the inverse of the following matrix.

42

23A

Find the Inverse Matrix.

28

16A

We use inverse matrices to solve Matrix Equations (Just like we use inverse operations to solve algebraic equations!!)

A B

Multiply both sides by A-1. Be sure to put A-1 first in order:

A-1· AX = A-1· BIX = A-1· B X = A-1· B

22

83

75

43X

Solve:

22

83

75

43X

Solve:

91

27

74

21X

To use inverse matrix on calc:

Enter Matrix name, then use x-1 button

Try solving the previous problem on your calculator now.

Find the inverse of the following on your calculator. Verify your results.

113

213

321