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What you will learn

1. What an identity matrix is2. How to find the inverse of a 2x2 and

other size matrices3. How to decode messages using inverse

matrices

Objective: 4.4 Identity and Inverse Matrices

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Identity Matrix

Any number multiplied by 1 is the same number (multiplicative identity). What is the “multiplicative identity” for a matrix?

An identity matrix for any matrix is the matrix that has 1’s on the main diagonal and 0’s elsewhere. e.g.

10

01

100

010

001Identity for 2x2

Identity for 3x3

Objective: 4.4 Identity and Inverse Matrices

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Inverse Matrices What happens when I multiply ½ by 2?

What do you suppose would happen if I multiplied any matrix by its inverse?

Let’s see: Matrix B is the inverse of Matrix A. Find AB:

We refer to the inverse of A as A-1.

25

13

35

12

Objective: 4.4 Identity and Inverse Matrices

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How Do You Get the Inverse of a Matrix?

Glad you asked!

The inverse of Matrix A= is:

Provided the determinant of A is not zero.

dc

ba

ac

bd

cbadac

bd

AA

111

Objective: 4.4 Identity and Inverse Matrices

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Example Find the inverse of A

=

24

13

ac

bd

cbadac

bd

AA

111

Objective: 4.4 Identity and Inverse Matrices

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On A Calculator Find the inverse of A

=

24

13

Objective: 4.4 Identity and Inverse Matrices

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Finding the Inverse of a 3x3 Matrix We will only do this on a calculator! Yeah!

Example: Find the inverse of A:

326

101

011

Objective: 4.4 Identity and Inverse Matrices

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Using Inverse Matrices Encoding and decoding messages. Number the letters A through Z, 1

through 26. Let 0 = a blank space.Example: Convert “Get Help” to a coded

set of 1 x 2 matrices.G E T _ H E L P

Objective: 4.4 Identity and Inverse Matrices

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Encoding Use the converted matrices from the previous

“get help” example. Use A = to encode the

matrices.

[ 7 5 ]

[20, 0]

[8, 5]

[12, 16]

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32

Objective: 4.4 Identity and Inverse Matrices

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The Encoded Message The encoded message should be:

9, 11, 40, 60, 11, 14, 8, 4

To decode, find the inverse of the coding matrix:

Multiply each of the 1 x 2 matrices by the inverse to decode the message.

21

32

Objective: 4.4 Identity and Inverse Matrices

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Decoding (continued) 9, 11, 40, 60, 11, 14, 8, 4 and inverse:

21

32

Objective: 4.4 Identity and Inverse Matrices

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You Try

Use the inverse of A = to decode the following:

-4, -1, -12, 7, -18, 18, 11, -20, -12, 7, 42, -63, -16, 16

32

11

Objective: 4.4 Identity and Inverse Matrices

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Homework Page 227, 14-18 even, 34, 36, 38, 44, 46