4.4 Identify and Inverse Matrices

Algebra 2

Learning Target

I can find and use inverse matrix.

Introduction

There are certain properties of real numbers that are related to special matrices. Remember that 1 is the identity for multiplication because 1 ∙ a = a ∙ 1 = a. The identity matrix is a square matrix that, when multiplied by another matrix, equals that same matrix.

With a 2 x 2 matrices, is the identity matrix

because

dc

ba

dc

ba

10

01

10

01

dc

ba

dc

ba

10

01

and

The identity matrix is symbolized by I. In any identity matrix, the principal diagonal extends from upper left to lower right and consists only of 1’s

Identity Matrix for Multiplication(The rule)

The identity matrix I for multiplication is a square matrix with a 1 for every element of the principal diagonal and a 0 in all other positions.

Ex. 1: Find I so that

148

123

148

123I

100

010

001

1

1

1

In order for you to multiply the matrices, remember that the number of columns of the first matrix must equal the number of rows in the second one.

The dimensions of the first matrix are 2 x 3. So I must have 3 rows. Since all identity matrices are square, it also has 3 columns. The principal diagonal contains 1’s. Complete the matrix with 0’s

The 3 x 3 identity matrix is

100

010

001

Next

Another property of real numbers is that any real numbers is that any real number except 0 has a multiplicative inverse. That is 1/a is the multiplicative inverse of a because a ∙ 1/a = 1/a ∙a = 1. Likewise, if matrix A has an inverse named A-1, then A ∙ A-1= A-1∙ A = I. The following example shows how a 2 x 2 matrix can be found.

Inverse form for a 2 x 2

dc

ba

ac

bd

bcad

1

For any matrix M, will have an inverse M-1

If an only if Then M-1 =

dc

ba

Ex. 3: If A = ,find A-1 and check your results

41

53

751241

53

31

54

7

11A

10

01

12533

2020512

7

1

41

53

31

54

7

1

Compute the value of the determinant.

Since the determinant does not equal 0, A-1 exists.

Check:

Assignment

Pg. 227 #6-24 Pg 233 12-30