Identity & Inverse Matrices

Identity

In other words, 5 * __= 5?

What does “identity” mean to you?

What is the multiplicative identity for the real numbers?

The identity for multiplication is 1 because anything multiplied by 1 will be itself.

Inverses

What does “inverse” mean to you?

What is the inverse of multiplication?

a * a-1= 1

In other words, 5 * ___=1?

What do we multiply by to get the identity?

Any number multiplied by its inverse will be the identity.

Identity MatrixIdentity Matrix The multiplicative identity for The multiplicative identity for matrices is a square matrix with matrices is a square matrix with ones on the main diagonal and ones on the main diagonal and

zeros everywhere else.zeros everywhere else.

10

01I

100

010

001

I

Identity Matrix

AI= A

Just like 5*1 = 5…

10

01

IA= A

178

132

1781

32

10

01

1781

32

1781

32

Or

Identity Matrix

A * A-1= I

A-1 *A = I

Any matrix multiplied by its inverse will be the identity matrix.

10

01I

2x2 Identity Matrix

100

010

001

I

3x3 Identity Matrix

Ex. 1 Determine whether A and B are inverses.

63

32A

3

21

12B

YES

Ex. 2 Determine whether A and B are inverses.

47

35A

57

34B

NO

The Inverse of a 2x2 Matrix

dc

baA

AA-1-1==1 d b

c aA

As long as ad-cb As long as ad-cb ==00

If ad-cd=0, then the matrix If ad-cd=0, then the matrix has no inverse!!!!has no inverse!!!!

1 d b

c aad bc

Ex. 3 Find A-1, if it exists.

75

32A

25

371A

25

37

1514

1AA-1-1==

Ex. 4 Find A-1, if it exists.

04

12A

2

11

4

10

24

10

4

1AA-1-1==

Ex. 5 Find A-1, if it exists.

01

62

43

A

Does not exist, because it’s not square.

Now let’s learn how to Now let’s learn how to use our calculator!!!use our calculator!!!

75

32A

Find the inverse!Find the inverse!

04

12A

Yes, now you can add, subtract, multiply, Yes, now you can add, subtract, multiply, and find the determinant in you calculator!!and find the determinant in you calculator!!

Solving Systems using Matrices and Inverses

Suppose ax = b

How do you solve for x?

We cannot divide matrices, but we can multiply by the inverse.

AX = BAA-1-1 AA-1-1

IX = AA-1-1B

X = AA-1-1B

Solving Matrix Equations

Solving a Matrix Equation

36

58

13

14X

Solve the matrix equation AX=B for the 2x2 matrix X

30

22X

X = AA-1-1B

Ex. Solve

22

83

75

43X

3421

4829X

Solving Systems Using Solving Systems Using Inverse MatricesInverse Matrices

5 2 3

4 2 4

x y

x y

Setting Up the Matrices

• Matrix A will be the coefficients of the system

• Matrix X will be the variables

• Matrix B will be constants (what the system of equations are equal to)

Matrix Equation

6

8

21

45

y

x

A linear system can be written as a A linear system can be written as a matrix equation AX=Bmatrix equation AX=B

Coefficient Coefficient matrixmatrix Variable Variable

matrixmatrix

Constant Constant matrixmatrix

5 4 8

1 2 6

x y

x y

6

8

21

45

y

x

5 4 8

1 2 6

x y

x y

Example 1

Example 2: Use matrices to solve the linear system

5 2 3

4 2 4

x y

x y

5 2

4 2

x

y

3

4

Find the inverse

(-1, 4)

1 1

52

2

3

4

x

y

Type in [A]-1 [B]

Example 3: Use matrices to solve the linear system4 2 8

2 12

x y

x y

Find the inverse

(4, 4).2 .2

.1 .4

8

12

x

y

4 2 8

1 2 12

x

y

Type in [A]-1 [B]

Example 4: Use matrices to solve the linear system

2 3

2 3 4

4 3 18

x y z

x y z

x y z

(-2, 3, 1)

18

4

3

134

312

211

z

y

xType in [A]-1 [B]

Example 5: Use matrices to solve the linear system

2 2

5 5

2 2 0

x z

x y z

x y z

(2, 3, -2)

2 0 1 2

5 1 1 5

1 2 2 0

x

y

z

Type in [A]-1 [B]

Let’s apply this…

You have $18 to spend for lunch during a 5 day school week. It costs you $1.50 to make lunch at home and $5 to buy lunch. How many times each week do you make a lunch at home?

5

1.5 5 18

x y

x y

1 1 5

1.5 5 18

x

y

Type in [A]-1 [B](2, 3)

You make lunch at home 2 times a week.

A word problem…!!

• A small corporation borrowed $1,500,000 to expand its product line. Some of the money was borrowed at 8%, some at 9% and some at 12%. How much was borrowed at each rate if the annual interest was $133,000 and the amount borrowed at 8% was 4 times the amount borrowed at 12%?

$800,000 at 8%$800,000 at 8%

$500,000 at 9%$500,000 at 9%

$200,000 at 12%$200,000 at 12%

Homework