MATRICES

BA301ENGINEERING

MATHEMATICS 3

JABATAN MATEMATIK,SAINS DAN KOMPUTERPOLITEKNIK TUANKU SULTANAH BAHIYAH

CHAPTER 4

MATRICES

INDEX

OBJECTIVES

DIMENSIONS

TYPES OF MATRICES

OPERATION ON MATRICES

- Addition

- Subtraction

- Multiplication

- Scalar Multiplication

SOLVING MATRICES PROBLEMS

TRANSPOSE OF MATRIX

WHAT IS A MATRIX

TUTORIAL A

SOLVING SIMULTANEOUS EQUATION

INVERSE MATRIX

CRAMER’S RULE

OBJECTIVES:

- Know characteristics of matrices

- Apply basic operations on matrices- Know the inverse matrices (up to 3X3)

- Solve simultaneous linear equations up to 3 variables

OBJECTIVES :-Know characteristics of matrices-Apply basic operations on matrices-Know the inverse matrices (up to 3X3)-Solve simultaneous linear equations up to 3 variables

What is a matrix ???

Matrix : A rectangular arrangement of numbers in rows

and columnsOrder/Dimensions/Size of a matrix is the number of the

rows and columnsEntries /Elements are the numbers in the matrix


Describing Dimensions of the Matrix

row

234

012

column

(# no. of row) x ( # no. of column)

DIMENSIONS OF THE MATRIX :

So……..This is a 2 X 3 matrix


A matrix with m rows and n columns is

called a matrix with dimensions m x n

Example : Find the dimensions of the matrix.

197

324)a

29

37

05

)b

381) c

512

348

1097

)d

2 X 3 3 X 2

1 X 3 3 X 3


PRACTICE :

Find the dimensions of the following matrix

5)a

7

1)b

23

49)c

1 X 1

2 X 1

2 X 2

TYPES OF

MATRICES


SQUARE MATRIX

No of rows = No of column

Example :-

23

45T 2 X 2

Square matrix

534

026

148

A 3 X 3

Square matrix


DIAGONAL MATRIX

All entries = 0 EXCEPT at least one Of the diagonal entries

Example :-

100

050

003

T

diagonal

001

050

300

T

Diagonal

Matrix


IDENTITY MATRIX

Diagonal entries = 1 All other entries = 0

Example :-

100

010

001

I

001

010

100

I

10

01I

01

10I

example


TRANSPOSE MATRIX

Element in row change into column and element in column change into rows

Example :-

181

657

423

B

164

852

173TB



OPERATION ON

MATRICES

ADDITION

SUBTRACTION

SCALAR MULTIPLICATION

TRANSPOSE OF MATRIX

MULTIPLICATION


ADDITION ON MATRICES

-Both matrix must have the same dimensions to do addition

operation

-To add matrices, we add the corresponding elements


Example :

45

12A

87

65B

Find A + B

BA

2 + 5

1 + 6 5 +

7 4+ 8

1212

77BA


CALCGEO

PRACTICE :

Find A + B

4532) BandAa

323

942

987

238) BandAb

77 BA

12104

1176BA


SUBTRACTION ON MATRICES

-Both matrix must have the same dimensions to do subtraction

operation-To subtract matrices, we subtract the corresponding

elements


Example :

45

12A

87

65B

Find A - B

BA 2 -

51 - 65 -

74- 8

42

53BA


GEO

PRACTICE :

Find A - B

4532) BandAa

323

942

987

238) BandAb

13 BA

6610

7110BA


SCALAR MULTIPLICATION ON MATRICES

297

613

452

k

kkk

kkk

kkk

297

613

452

Multiply each element inside the matrix by k

OBJECTIVES :-Know

characteristics of matrices

-Apply basic operations on matrices

-Know the inverse

matrices (up to 3X3)

-Solve simultaneous

linear equations up to

3 variables

Example :

63

425Solve

5 X

25 X -

45 X -3

5 X 6

3015

2010

OBJECTIVES :-Know



-Know the inverse


-Solve simultaneous


3 variables

GEO

PRACTICE :

Solve the following :

1293

4) a

1612

987

2385)b

454035

101540

OBJECTIVES :-Know



-Know the inverse


-Solve simultaneous


3 variables

MULTIPLICATION ON MATRICES

Look at the dimension of matrices A and B to do MULTIPLICATIONS

m x n n x pmust be same

size of product

If the numbers of columns of A does NOT EQUAL the number of rows of B,

then… A X B is

UNDEFINED..!!


Example :

743

152A

8

3

2

Band

Find A X B.

8

3

2

743

152BXA

dimensions?? 2 X 3 3 X 1

same

Size of product = 2 X 1

VERY IMPORTANT!!


AXB (2 x 2) + (5x 3) + (1x 8)

(3x2) + (4x3) + (7x8)

56126

8154AXB

74

27AXB

8

3

2

743

152BXA


GEO

PRACTICE :

..1264

578

25

12

39

) AXBFindBandAa

..

7

5

6

458) PXQFindQandPb

22820

98160AXB

5PXQ


..

2025

753

968

354

231

867

) MXNFindNandMc

614332

701327

1815678

MXN

2

8

53

14) UandTd

14

34MXN


SOLVING MATRICES PROBLEMSExample :

yandxFindx

y

x.

130

1219

106

68

128

2533

3

9x-15 6

24 6y+3

130

1219

106

68x

13630

121517

y

x

130

1219


17 x - 15 =19 17 x = 15 +

19 17 x = 34x = 34/17

x = 2

6 y + 13 = 1 6 y = 13 -

1 6y = 12

y = 12/6 y =

2


TRANSPOSE OF MATRIX

Tips :

BECOME

column

row


TRANSPOSE OF MATRIX

Example :

TA

..

25

12

39TAFindAGiven

9 2 5-3 1 -2


GEO

PRACTICE :

1264

578) Ba

Find transpose of the following matrices :

125

67

48TB

6

7

8

) Pb

678TP


TUTORIAL A

CBdABDcCAbBAa

Find

DCBA

))()))

:

.84

53

5

3

62

43

59

83.1

BAbABa

Find

FindBandAIf

))

:

:.62

40

59

61.2

..

3

5

2

634

059

762

.3 MNFindNandMIf

Answer

Answer

Answer


2))

:.

139

357

761

45

59

61

.5

BbBAa

FindBandAIf

TT BAbBAa

FindBandAGiven

)())

:.248

869

34

55

62

.6

BAb

BAa

FindBandAmatrixGiven

)

)

:.

39

48

61

30

59

62

.4

Answer

Answer

Answer


).1 asolutionnoCBd

ABDc

solutionnoCAb

BAa

Find

DCBA

)

37

13)()

)

17

126)

:

.84

53

5

3

62

43

59

83.1

TUTORIAL A


4256

2036)

610

3212)

:

:.62

40

59

61.2

BAb

ABa

Find

FindBandAIf

TUTORIAL A


41

43

55

..

839

248

761

3

5

2

634

059

762

.3

MN

MNFindPandNMIf

TUTORIAL A


09

11

01

)

69

917

123

)

:.

39

48

61

30

59

62

.4

BAb

BAa

FindBandAmatrixGiven

TUTORIAL A


737239

677669

3257106

)

7341

7967

6490

)

:.

139

357

761

45

59

61

.5

2Bb

BAa

FindBandAIf

TUTORIAL A


5914

121111)

512

911

1411

)

:.248

869

34

55

62

.6

BAb

BAa

FindBandAGiven

T

T

TUTORIAL A


INVERSE OF MATRIX 2 X 2

3 X 3Formula :

adjAA

A 11

..65

45 1

AFindA

)4(5)6(5 A

2030 A

10A

Example :

1. Determinant A

GEO

2 X 2


2. Minor A

65

45A

65

45M What you can

see??611 m 512 m

421 m 522 m

54

56Minor


3. Cofactor A

54

56Minor

54

56Cofactor

4. Adjoint A

55

46AAdj

adjAA

A 11

55

46

10

11A

5. Inverse A

2

1

2

15

2

5

3

10

1

GEO


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INVERSE OF MATRIXFormula :

Example :

.sin.

317

235

1121 MatrixInversegubyAFindA

1. Determinant A

77317

35235

12112

A

332 A 721 751 137 221 153

351418 A 15421

33A

3 X 3

adjAA

A 11


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2. Minor

317

235

112

A

11

2133

31

23

11

11

11

m

m

m

1

2735

37

25

12

12

12

m

m

m

26

3715

17

35

13

13

13

m

m

m

2

1131

31

11

21

21

21

m

m

m

1

7132

37

12

22

22

22

m

m

m

5

1712

17

12

23

23

23

m

m

m

5

3121

23

11

31

31

31

m

m

m

1

5122

25

12

32

32

32

m

m

m

1133

513233

35

1233

m

m

m


1115

512

26111

M

3. Cofactor

1115

512

26111

K

1115

512

26111

K

4. Adjoint

11526

111

5211

Aadj


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5. Inverse A

11526

111

5211

33

11A

3

1

33

5

33

2633

1

33

1

33

133

5

33

2

3

1

33

11A


Cramer’s Rule Formula :

A

Az

A

Ay

A

Ax zyx

SOLVING SIMULTANEOUS EQUATION

INVERSE MATRIX CRAMER’S RULE

Inverse Matrix Formula :

adjAA

A 11


CRAMER’S RULEExample :

.'sin,

8482

5926

475

RulesCramergubyzandyxFind

zyx

zyx

zyx

82482

26926

15715

A

1. Determinant A

A 425 291 867 722 598 164

]2436028[3361840 A

2A


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2. Determinant Ax

8848

2592

1471

xA458

562

xA 424 891 857 728 498 154

]20288112[2807232 xA

44xA

3. Determinant Ay

8242

5696

4575

xA458

-1-2 8

yA 455 294 867 752 598 464

]9636070[33672100 yA

26yA












4. Determinant Az

82882

26526

15415

zA

zA 825 864 251 422 558 168

]4820016[1921080 zA

34zA x22

2

44x 13

2

26

y 17

2

34z







MATRICES

Technology

inverse matrices

characteristics of matrices

matrices addition

simultaneous linear

basic operations

matrix byobjectives

column dimensions

index objectives