MATRICES

Jeffrey BivinLake Zurich High School

jeff.bivin@lz95.org

Last Updated: October 12, 2005

Inverses and Identities

5x = 351

51

531 x53x

inversestivemultiplicaareand 551

identitytivemultiplicatheis1

Jeff Bivin -- LZHS

1001

Now with Matrices

4321

4321

This is theIdentity Matrix

for 2 x 2 Matrices

Jeff Bivin -- LZHS

Let’s look at another example

5387

1001

53

87

Jeff Bivin -- LZHS

New Question

4321

1001

What do we multiply a matrix

by to get the Identity?

Jeff Bivin -- LZHS

The Inverse of a 2x2 Matrix

dcba

dcba1

acbd

0dcba

Jeff Bivin -- LZHS

The Inverse of a 2x2 Matrix

4321

43211

1324

1324

64121

21

23

12

Jeff Bivin -- LZHS

The Inverse of a 2x2 Matrix

2134

2134

1

41

32

41

32381111

114

111

113

112

Jeff Bivin -- LZHS

:' usesLet

3125

4321X

BAX BAX 1A

BAXI 1

1A

BAX 1This is our Formula!

3125

1324

43211X21

9141418

21X

29779

XJeff Bivin -- LZHS

:' usesLet

3427

5431

X

BXA BXA 1A

1 ABIX

1A

1 ABXThis is our Formula!

14

353427

5431

1X

15322343

71X

715

732

723

743

71

1435

3427

71X

Jeff Bivin -- LZHS

:' usesLet

6314

4132

3251X

CBAX

)( BCAX 1A BCAXI 1

1A

BCAX 1

This is our Formula!

41

326314

1253

3251

1X71

2442

1253

71X

68226

71X

BCAX

76

78

72

726

Jeff Bivin -- LZHS

Are the two Matrices Inverses?

8532

2538

1001

Jeff Bivin -- LZHS

The product of inverse matrices

is the identity matrix.

Identity, therefore, INVERSEMatrices

Are the two Matrices Inverses?

4123

3124

100010

Jeff Bivin -- LZHS

The product of inverse matrices

is the identity matrix.

Not the Identity,

therefore, NOT

INVERSEMatrices

4623

Jeff Bivin -- LZHS

Does the Matrix have an Inverse?

Let’s review the definition of the Inverse of a

2x2 Matrix

The Inverse of a 2x2 Matrix

dcba

dcba1

acbd

0dcba

Jeff Bivin -- LZHS

4623

Find the determinant!

4623

Jeff Bivin -- LZHS

01212

Therefore, NO inverse!

Does the Matrix have an Inverse?

14427

Find the determinant!

14427

Jeff Bivin -- LZHS

90898

Therefore, an inverse exists!

Does the Matrix have an Inverse?

987654321

Find the determinant!

Jeff Bivin -- LZHS

Therefore, NO inverse!

Does the Matrix have an Inverse?

1 2 3 1 24 5 6 4 57 8 9 7 8

1•5•9 + 2•6•7 + 3•4•8 - 7•5•3 - 8•6•1 - 9•4•2

45 + 84 + 96 - 105 - 48 - 72

243142231

Find the determinant!

Jeff Bivin -- LZHS

Does the Matrix have an Inverse?

1 3 2 1 32 4 1 2 43 4 2 3 4

1•4•2 + 3•1•3 + 2•2•4 - 3•4•2 - 4•1•1 - 2•2•3

8 + 9 + 16 - 24 - 4 - 12

Therefore, an inverse exists!

:' usesLet

11

75423

yx

BAU BAU 1A

BAUI 1

1A

BAU 1This is our Formula!

117

3425

54231U 231

557

231U

2352357

Jeff Bivin -- LZHS

3x + 2y = 74x - 5y = 11

Solve the system using inverse matrices

235

2357 , solution

:' usesLet

19

2342

yx

BAU BAU 1A

BAUI 1

1A

BAU 1This is our Formula!

1

92342

2342

1U 81

2514

81U

82547

Jeff Bivin -- LZHS

2x - 4y = 93x - 2y = 1

Solve the system using inverse matrices

825

47 , solution