IDENTITY AND INVERSE MATRICES

Lesson 4.7

Identity Matrix:

If you multiply a matrix by an identity matrix (I) the result is the same as the original matrix.

If Matrix A is a square matrix then it is also commutative. A I I A

A I A

_ _

_ _

a b a b

c d c d

1 0

0 1

a b a b

c d c d

0 0 00 0

0 0 0 0 or 0

11

1 11

1 11

0 or 0 0 0 0

0

2x2 3x3

00 1

4x4

0 0

•Ones on the principal diagonal and all the rest are zeros.

Inverse Matrix

If two inverse matrices are multiplied the resulting product is the Identity Matrix.

1A A I ? ? 1 0

? ? 0 1

a b

c d

For a 2x2 inverse matrix

1If then 1

det

d bM

c a

a bM

c d M

1 9 41

5 218 2

2 4If , then

9 05MM

For example:

921

52

29 41 or

15 22M

What about inverse matrices of other sizes?

Use calculator for now. If you go on in math, you will learn that

in Linear Algebra.

Using Inverse Matrices for Encryption and Decryption

Basic phrase: Bulldogs Translate to a numerical string where A=1,

B=2, etc.

2 21 12 12 4 15 7 19 Put this string into a matrix,

maybe a 4x2 2 21

12 12

4 15

7 19

[ 0 N 14

A 1 O 15

B 2 P 16

C 3 Q 17

D 4 R 18

E 5 S 19

F 6 T 20

G 7 U 21

H 8 V 22

I 9 W 23

J 10 X 24

K 11 Y 25

L 12 Z 26

M 13

Multiply this matrix by another random matrix, in this case, if we do right hand multiplication, a simple 2x2 would work.

For example. 2 1

1 3

2 21

12 12 2 1

4 15 1 3

7 19

17 65

12 48results in

7 49

5 64

This matrix is converted back to a string of numbers.

-17 65 12 48 -7 49 5 34

17 65

12 48

7 49

5 64

To get back to the original message, the receiver would then use the inverse of our encryption matrix to get it back.

3 17 7

1 27 7

17 65

12 48

7 49

5 64

2 21

12 12 to get back to

4 15

7 19

[ 0 N 14

A 1 O 15

B 2 P 16

C 3 Q 17

D 4 R 18

E 5 S 19

F 6 T 20

G 7 U 21

H 8 V 22

I 9 W 23

J 10 X 24

K 11 Y 25

L 12 Z 26

M 13

2 21

12 12

4 15

7 19

Which translates back to:

Bulldogs

Work Time 10 minutes.

Using Matrix Equations to solve systems of equations4 2 17

3 4 10

x y

x y

Can be converted to a matrix equation, where the variables are in their own matrix, like this:

4 2 17

3 4 10

x

y

xA B

y

Coefficient matrix ∙ variable matrix = constant matrix

4 2 17

3 4 10

x

y

xA B

y

Use inverse matrices to solve:

Multiply both sides by A-1 because A-1A gives the identity.

1 1

I

xA B

yA A

12

4x

yI

Try this one:

•First, turn it into a matrix equation

•Then label A and B

•Then multiply both sides by the inverse of A. (since it is a 3x3, use calculator to do calculation.)

•Make sure you check your solution.

5 7 3 2

2 5 5 21

4 2 13 5

x y z

x y z

x y z

5 7 3 2

2 5 5 21

4 2 13 5

x

y

z

1 1

x

A A y A B

z

A B

3

2

1

x

y

z

(3, 2,1)