Matrices & Systems of Linear Equations. Special Matrices.

Matrices & Systems of Linear Equations

Special Matrices

000

000

000

,0

0

0000,00

00

:

3321

1422

OO

OO

Examples

MatrixZero

0214

4513

8102

8101

453

912

601

,25

01,5

:Examples

MatrixnbynAn

MatrixSquare

Special Matrices

8000

0500

0070

0001

400

010

001

,20

01,5

:Examples

zeroesarediogonal

maintheonnotarethatentriesall

ifmatrixsquareA

MatrixDiagonal

1000

0100

0010

0001

100

010

001

,10

01,1

:

44

332211

I

III

Examples

onesarediogonal

maintheonarethatentriesall

ifmatrixdiagonalisA

MatrixIdentity

Equality of Matrices

Two matrices are said

to be equal if they have

the same size and their

corresponding entries

are equal

45

21,

43

21

?

584

573

062

951

,

5509

8765

4321

?

equalmatricesfollowingtheAre

equalmatricesfollowingtheAre

Equality of Matrices

Use the given equality

to find x, y and z

52343

21

.2

509

8735

4221

5509

8765

4321

.1

25

z

yx

z

y

x

Matrix Addition and SubtractionExample (1)

12915

1296

471

391887

664524

135201

318

642

150

987

654

321

Matrix Addition and SubtractionExample (2)

671

012

231

391887

664524

135201

318

642

150

987

654

321

Multiplication of a Matrix by a Scalar

21176

666

512

318

642

150

181614

12108

642

318

642

150

)1(

987

654

321

2

)2(

45535

30020

15105

)9(5)1(5)7(5

)6(5)0(5)4(5

)3(5)2(5)1(5

917

604

321

5

)1(

Example

Example

Matrix Multiplication(n by m) Matrix X (m by k) Matrix

The number of columns of the matrix on the left

= number of rows of the matrix on the right

The result is a (n by k) Matrix

Matrix Multiplication3x3 X 3x3

332211332211332211

332211332211332211

332211332211332211

333

222

111

321

321

321

zczczcycycycxcxcxc

zbzbzbybybybxbxbxb

zazazayayayaxaxaxa

zyx

zyx

zyx

ccc

bbb

aaa

Matrix Multiplication1x3 X 3x3→ 1x3

32211332211332211

333

222

111

321

azazayayayaxaxaxa

zyx

zyx

zyx

aaa

Example (1)

323

422

241

)2)(1()1)(1()0)(2()1)(1()1)(1()1)(2()1)(1()0)(1()2)(2(

)2)(2()1)(0()0)(0()1)(2()1)(0()1)(0()1)(2()0)(0()2)(0(

)2)(1()1(4)0(1)1)(1()1(4)1(1)1)(1()0(4)2(1

211

110

012

112

200

141

Example (2)(1X3) X (3X3) → 1X3

52411

)4(4)10(3)3(2)3(4)2(3)1(2)2(4)1(3)0(2

432

1021

310

432

Example (3)(3X1) X (1X2) → 3X2

00

76

3530

)7(0)6(0

)7(1)6(1

)7(5)6(5

76

0

1

5

Example (4)

37164

1110

043

540

311

121

540

311

121

540

311

1212

Transpose of Matrix

863

752

041

870

654

321

)1(

333

222

111

321

321

321

T

T

Example

cba

cba

cba

A

ccc

bbb

aaa

A

30273

251214

9102

200

020

002

32273

251014

9104

100

010

001

2

24210

181512

963

863

752

041

2

870

654

321

3

870

654

321

)2(

3I

Example

T

870

654

321

870

654

321

)3(

TT

Example

Properties of the Transpose

TTT

TT

ABAB

AA

)(.2

)(.1

Matrix ReductionDefinitions (1)

1. Zero Row: A row consisting entirely of zeros

2. Nonzero Row: A row having at least one nonzero entry

3. Leading Entry of a row: The first nonzero entry of a row.

Matrix ReductionDefinitions (2)

Reduced Matrix: A matrix satisfying the following:

1. All zero rows, if any, are at the bottom of the matrix

2. The leading entry of a row is 1

3. All other entries in the column in which the leading entry is located are zeros.

4. A leading entry in a row is to the right of a leading entry in any row above it.

Examples of Reduced Matrices

000

710

501

.3

000

010

001

.2

100

010

001

.1

Examples matrices that are not reduced

.

2

000

001

010

.2

12

100

060

001

.1

itaboverowtheinentry

leadingtheofrightthetonotisrowinentryleadingThe

notisrowinentryleadingThe

)3(

,12

000

210

031

.4

.

2

010

000

001

.3

columntheintheNotice

zerosarelocatedisitwhichin

columntheinentriesotherallnotbut

isrowinentryleadingThe

matrixtheofbottomtheatnotbut

rowzeroaisRow

Elementary Row Operations

1. Interchanging two rows

2. Replacing a row by a nonzero multiple of itself

3. Replacing a row by the sum of that row and a nonzero multiple of another row.

Interchanging Rows

225

320

1263

225

1263

32021 RR

Replacing a row by a nonzero multiple of itself

225

320

421

225

320

12631

3

1R

Replacing a row by the sum of that row and a nonzero multiple of another row

22120

320

421

225

320

421)5( 13 RR

Augmented Matrix Representing a System of linear Equations

7

3

7

225

1263

320

:

7225

31263

732

:

matrixaugmentedthebydrepresenteIs

zyx

zyx

zy

equationslinearofsystemThe

29

)(

3

4

10

01

9

4

20

01

9

5

20

21

6

5

43

21

:

6

5

43

21

:

Re

)1(

221

21

12

R

RR

RR

Solution

operations

rowbasicapplyingbymatrixaugmentedfollowing

theofrighttheonmatrixtwobytwotheduce

Example

Solving a System of Linear Equations by Reducing its Augmented Matrix

Using Row Operations

29

29

&4

:

4

10

01

:

6

5

43

21

:

:

:

643

52

:

)1(

yx

thatconcludeWe

matrixtheatarrivingmatrixthisreduceWe

operations

matrixaugmentedtheconstructWe

Solution

yx

yx

equationslinearofsystemfollowingtheSolve

Example

7

3

7

225

1263

320

:

7225

31263

732

:

)2(

matrixaugmenteditsreducingby

zyx

zyx

zy


Example

Solution

2

8

22120

10

701

2

1

22120

10

421

2

7

1

22120

320

421

7

7

1

225

320

421

7

7

3

225

320

1263

7

3

7

225

1263

320

27

23

)2(

27

23

2

1

)5(3

1

212

131

21

RRR

RRR

RR

1

2

1

100

010

001

1

1

100

10

001

1

8

100

10

701

40

8

4000

10

701

2

8

22120

10

701

)(

27

23

7

27

23

40

1

27

23

12

27

23

323

2

313

23

RR

RRR

RR

Solution of the System

1

2

1

,

1

2

1

100

010

001

z

y

x

Thus

The Idea behind the Reduction Method

7

3

7

225

1263

320

:matrixaugmentedThe

7225

31263

732

:

zyx

zyx

zy

equationslinearofsystemThe

Interchanging the First & the Second Row

7

7

3

225

320

1263

7

3

7

225

1263

32021 RR

7225

732

31263

7225

31263

732

:)2()1(

zyx

zy

zyx

zyx

zyx

zy

EqandEqBetweem

placesthengtIntercangi

Multiplying the first Equation by 1/3

7

7

1

225

320

421

7

7

3

225

320

1263

13

1R

7225

732

142

7225

732

31263

zyx

zy

zyx

zyx

zy

zyx

Subtracting from the Third Equation 5 times the First Equation

2

7

1

22120

320

421

7

7

1

225

320

421

)5( 13 RR

22212

732

142

7225

732

142

zy

zy

zyx

zyx

zy

zyx

Subtracting from the First Equation 2 times the Second Equation

2

8

22120

10

701

2

1

22120

10

421

27

23

)2(

27

23

21 RR

222122

7

2

3

87

222122

7

2

3

142

zy

zy

zx

zy

zy

zyx

Adding to the Third Equation 12 times the Second Equation

40

8

4000

10

701

2

8

22120

10

701

27

23

12

27

23

23 RR

40402

7

2

3

87

222122

7

2

3

87

z

zy

zx

zy

zy

zx

Dividing the Third Equation by 40

1

8

100

10

701

40

8

4000

10

701

27

23

40

1

27

23

3R

12

7

2

3

87

40402

7

2

3

87

z

zy

zx

z

zy

zx

Adding to the First Equation 7 times the third Equation

1

1

100

10

001

1

8

100

10

701

27

23

7

27

23

31 RR

12

7

2

3

1

12

7

2

3

87

z

zy

x

z

zy

zx

Subtracting from the Second Equation 3/2 times the third Equation

1

2

1

12

7

2

3

1

z

y

x

z

zy

x

1

2

1

100

010

001

1

1

100

10

001

)(

27

23

323

2 RR

Systems with infinitely many Solutions

:

23

:,

23

:,

?)(

sec

642

32

:

:)1(

tableoppositetheinshownAs

rx

getwernumberrealanybeylettingBy

yx

equationoneonlyhaveweThus

Whyfirsttheas

sametheisequationondthethatNotice

yx

yx

systemfollowingtheSolve

Example

y = rx=3-2r

03

-15

11

10-17

yx

yx

Thus


matrixaugmentedtheconsidersLet

RR

23

00

32

:

0

3

00

21

:

6

3

42

21

:'

12 )2(

Systems with infinitely many Solutions

:

3:,

302

:),1(

033

:),3()1(

,

)3(

),2()1(

04

052

02

:

:)2(

tableoppositetheinshownAs

ryandrxgetwernumberrealanybezlettingThus

zxzzx

getweEqinthatngSubstituti

zyzy

getweEqfromEqgSubtractin

equationstindependentwoonlyhaveweThus

Eq

getweEqfromEqgsubtractinwhenthatNotice

zyx

zyx

zyx


Example

z=rx=-3ry=-r

000

1-3-1

-103010

1/3-1-1/3

ryandrx

getwernumberrealanybezlettingBy

zyandzx

zyandzx

usgivesrowsotherThe

ormationanycontributenotdoesrowlastThe

matrixtheatarrivingmatrixthisreducesLet


3

:,

3

003

:

00:inf

0

0

0

000

110

301

:'

0

0

0

411

512

121

:'

Details of reduction

0

0

0

000

110

301

0

0

0

000

411

301

0

0

0

000

411

903

0

0

0

000

411

121

0

0

0

411

411

121

0

0

0

411

512

121

)(

3

1

)2()(

)(

12

1

2123

12

RR

R

RRRR

RR

Systems with no Solution

solutionnohassystemtheThus

equationsbothsatisfyingyandxnumbersnoarethereThus

statementimpossiblethegetwefromEqEqgsubtractinBy

yxthatclaimsEqfirsttheBut

yx

bygMultiplyinnumberthebyEqontheDividing

WhyfirstthescontradictequationondthethatNotice

yx

yx


Example

10

:),3()1(

32:

)3(42

:)2

1(2sec

?)(sec

842

32

:

:)1(

impossibleisWhich

yx

Thus



RR

20

32

:

2

3

00

21

:

8

3

42

21

:'

12 )2(

slutionnohassystemthe

systemtheofequations

threetheallsaisfyingzandyxnumbersrealnoarethereThus

statementimpossiblethegetweEqfromEqgSubtractin

zythatstateswhichEqscontradictequationThis

zy

getweEqfromEqgsubtractinwhenthatNotice

zyx

zy

zyx


Example

.

,,

20

:),4()2(

32:),2(

)4(52

),1()3(

12

32

642

:

:)2(

10

02

0

)(

1

0

0

000

210

001

:'

1

3

6

211

210

421

:'

zy

x

solutionnohaswhichsystemthetoscorrespondThis

matrixtheatarrivingmatrixthisreducesLet


Details of the reduction

1

0

0

000

210

001

1

3

0

000

210

001

2

3

0

000

210

001

5

3

0

210

210

001

5

3

6

210

210

421

1

3

6

211

210

421

)3(

)2

1(

)()2(

)(

32

3

2321

13

RR

R

RRRR

RR

solutionnohaswhich

zyx

zy

systemthetodscorrresponWhich

approachbetterA

RR

RR

12

32

20

:

1

3

2

211

210

000

1

3

5

211

210

210

1

3

6

211

210

421

:

)(

)(

21

31

Finding the Inverse of an nXn square Matrix A

1. Adjoin the In identity matrix to obtain the Augmented matrix [A| In ]

2. Reduce [A| In ] to [In | B ] if possible

Then

B = A-1

Example (1)

1

115

014

001

100

920

201

101

014

001

820

920

201

100

014

001

1021

920

201

100

010

001

1021

124

201

:

1021

124

201

32

31

21

)(

)(

)4(

1

RR

RR

RR

Solution

AFindALet

115

4

229

,

115

4

229

100

010

001

115

9841

229

100

020

001

115

014

229

100

920

001

29

2411

29

241

)9(

)2(

221

23

13

A

Thus

R

RR

RR

Example (2)

inversenohasAThus

Solution

AFindALet

RR

RR

RR

111

012

001

000

430

321

103

012

001

430

430

321

100

012

001

533

430

321

100

010

001

533

212

321

:

533

212

321

32

31

21

)(

)3(

)2(

1

inversenohasAThus

WayAnother

RR

RR

111

011

001

000

533

321

100

011

001

533

533

321

100

010

001

533

212

321

:

32

21

)(

)(

Inverse MatrixThe formula for the inverse of a 2X2 Matrix

AAIAA

thatCheck

ac

bd

AA

ThenAIf

bcadA

dc

baA

Let

12

1

1

:

det

1

:,0det

det

&

93

52

3

21

3

53

10

01

3

21

3

53

93

52

3

21

3

53

23

59

3

1

23

59

det

1

03151893

52det

93

52:

:

1

Checking

AA

A

ALeT

Example

Using the Inverse Matrixto Solve System of Linear Equations

12

:

,

1

2

)15)(3

2()9)(1(

)15)(3

5()9(3

15

9

3

21

3

53

93

52

3

21

3

53

15

9

93

52

15

9

93

52

:Re

1593

952

:

2

yandx

isSolutionThe

Thus

y

x

y

xI

y

x

y

x

yx

yx

formmatrixinwriting

yx

yx


Problem

101

124

113

212

123

112

:

122

223

12

:

1

thatgivenIf

zyx

zyx

zyx


21,2

:

,

2

1

2

2

1

2

1

2

1

101

124

113

212

123

112

101

124

113

1

2

1

212

123

112

1

2

1

22

23

12

:Re

122

223

12

3

zandyx

isSolutionThe

Thus

z

y

x

z

y

x

I

z

y

x

z

y

x

zyx

zyx

zyx

formmatrixinwriting

zyx

zyx

zyx

Homework

6.1 Examples: 3 Exercises: 17 - 20

6.2 Examples: 1, 2, 4, 5 and 6 Exercises: odd numbered:1—17 and 25,29,35,37,39,41

6.3 Examples: 1, 2, 3, 4, 5, 7, 10, 11, 12 and 13. Exercises:19, 21, 23, 25, 27, 31, 33, 37, 51, 53, 57, 59, 61

6.5 Examples: 2, 3.a and 4. See given exercises.

6.6 Examples: 1 - 6 Exercises: odd numbered:1—15, 21, 27, 29, 35 and 37.

1.

Matrices & Systems of Linear Equations. Special Matrices.

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Transcript of Matrices & Systems of Linear Equations. Special Matrices.