Warm-Up

164

524

50

41

26

35

41

30

26

12

39

3) Find the determinant by hand.

145

41310

250

4) Find the determinant using your calculator.

1) Multiply. Show work.

2) Multiply. Show work.

Daily Quiz – you may not use your notes.

122

433

692

2 3 4

3) Find the determinant by hand.

4) Find the determinant using your calculator.

1) Multiply. Show work.

2) Multiply. Show work.

85

212

1428

6010

10415

12

01

10

23

33

4.4 Identity & Inverse 4.4 Identity & Inverse MatricesMatrices

InversesWhat does “inverse” mean to you?

What is the inverse of multiplication?

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.

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 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

Identity Matrix

AI= A

Just like 5*1 = 5…

10

01

IA= A

178

132

1781

32

10

01

1781

32

1781

32

Or

Ex. 1 Determine whether A and B are inverses.

24

13A

2

32

2

11

B

Find out if AB = I

If so, then yes they’re inverses.

If not, then no they’re not inverses.

Use your calculator.

YES

Ex. 2 Determine whether A and B are inverses.

63

32A

3

21

12B

YES

Ex. 3 Determine whether A and B are inverses.

47

35A

57

34B

NO

The Inverse of a 2x2 Matrix

dc

baA

AA-1-1==

ac

bd

bcadac

bd

A

11

As long as ad-cb As long as ad-cb ==00If ad-cd=0, then the matrix has no If ad-cd=0, then the matrix has no

inverse!!!!inverse!!!!

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

75

32A

25

371A

25

37

1514

1AA-1-1==

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

04

12A

2

11

4

10

24

10

4

1AA-1-1==

Ex. 6 Find A-1, if it exists.

01

62

43

A

Does not exist, because it’s not square.

Ex. 7 Find A-1, if it exists.

326

101

011

A

142

133

1321A

Solving Systems using Solving Systems using Matrices and InversesMatrices 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

36

58

13

14X

Step 1: find AStep 1: find A-1-1

43

111

43

11

34

1AA-1-1==

Step 2: Multiply both sides of the equationStep 2: Multiply both sides of the equationBy ABy A-1-1 on the left on the left

36

58

43

11

13

14

43

11X

36

58

43

11

13

14

43

11X

12152424

3568

431212

1134X

30

22

10

01X

30

22X

You can check theYou can check theSolution by multSolution by mult

AX to see if you get BAX to see if you get B

Ex. Solve

22

83

75

43X

3421

4829X

Assignment