Section 4.5

Identity and InverseMatrices

1 0 0 0 1 00 0 1

1 0 0 0 1 00 0 1

Def: A constant matrix with 1’s on the diagonal and 0’s everywhere else is called the identity matrix

Multiply

-2 4 7 5 -3 6-8 2 -1

• = (-2 + 0 + 0) (0 + 4 + 0)(0 + 0 -7)

(5 + 0 + 0)(0 -3 + 0)(0 + 0 + 6)

(-8 + 0 + 0)(0 + 2 + 0)(0 + 0 -1)

= -2 4 7 5 -3 6-8 2 -1

You get the same matrix

If the Identity matrix is designated as I then for anysquare matrix “A”:

A•I = A also I•A = A

commutative

The notation for inverse matrix is:

Notation: (Inverse of matrix M)

A-1

For matrices, the -1

is not an exponent

Def: When you multiply a matrix by its inverse matrix the product is the Identity Matrix

A•A-1 = I

12

35•

52

3 1=

10

01

54 x4

154

4

1 x 14

Def: Any matrix will have an inverse If and only if

and ac

bd

bcadA

11

dc

baA

Note: not all matrices have inverses

0dc

ba

1.) Use your calculator to find the inverse matrix for:

32

47A

13

7

13

213

4

13

31A

http://www.youtube.com/watch?v=_fFj4NbLcTU

2.) Use your calculator to find the inverse matrix for:

465

453

122

A

427

538

3241A

3.) Determine whether the following is true or false:

I

31

32

32

31

12

21

Using your calculator

Enter the matrix into your calculator2nd Matrix

EditAt this point just follow the prompts

Quit2nd Return to the home screen

2nd Matrix

Arrow down to the matrixyou defined in Step 1 above

Enter

Step

Call out the matrix to your home screen

Description

1

Key-strokes

2

3

Using your calculator

Find the inverse of your matrixx-1

The inverse should nowappear on your home screen

Step Description

4

Key-strokes

Use math, enter, enter to change decimals To fractions

Homework

Page 217Problems: 11- 19, 21-29(odd)