warm-up

Warm-Up

164524

504126

3541

3026

12

39

3) Find the determinant by hand.

14541310250

4) Find the determinant using your calculator.

1) Multiply. Show work.

2) Multiply. Show work.

Daily Quiz – you may not use your notes.

122433692

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

1428601010415

1201

102333

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= IA-1 *A = I

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

1001

I

2x2 Identity Matrix

100010001

I

3x3 Identity Matrix

Identity Matrix

AI= A

Just like 5*1 = 5…

1001

IA= A

178

132

1781

32

1001

1781

32

1781

32Or

Ex. 1 Determine whether A and B are inverses.

2413

A

232211

B

Find out if AB = IIf 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.

6332

A

32112

B

YES

Ex. 3 Determine whether A and B are inverses.

4735

A

5734

B

NO

The Inverse of a 2x2 Matrix

dcba

A

AA-1-1==

acbd

bcadacbd

A11

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.

7532

A

25371A

25

3715141

AA-1-1==

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

0412

A

211410

2410

41

AA-1-1==

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

016243

A

Does not exist, because it’s not square.

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

326101011

A

142133132

1A

Solving Systems using Solving Systems using Matrices and InversesMatrices and Inverses

Suppose ax = bHow 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-1-1BX = AA-1-1B

Solving Matrix Equations

Solving a Matrix Equation

3658

1314X

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

3658

1314X

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

43

111

4311

341AA-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

3658

4311

1314

4311

X

3658

4311

1314

4311

X

121524243568

4312121134X

3022

1001X

3022

XYou can check theYou can check the

Solution by multSolution by multAX to see if you get BAX to see if you get B

Ex. Solve

2283

7543X

34214829

X

Assignment