Matrices

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

MatrixnXnAn

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

Special Matrices

6000

1700

3960

4581

700

510

412

,30

12

:

44

3322

I

II

Examples

zerosarediogonal

mainthebelowarethatentriesall

ifmatrixsquareA

MatrixDiogonalUpper

6129

1731

0068

0004

758

016

002

,31

02

:

44

3322

I

II

Examples

zerosarediogonal

maintheabovearethatentriesall

ifmatrixdiagonalA

MatrixDiogonalLower

Matrix Addition and SubtractionExample

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

42411

)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)

2

2

37164

1110

043

540

311

121

540

311

121

540

311

121

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 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

zerosarelocatedisitwhichincolumnthein

entriesotherallthebutisrowinentryleadingThe

matrixtheofottomtheatnotbutrowzeroaisRow

,12

000

210

031

.4

.2

010

000

001

.3

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

theofrihgttheonmatrixtwobytwothetheduce

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

equationslinearofsystemfollowingtheSolve

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