Lecture 2 of 11 (Chap 5, Matrices)

8/12/2019 Lecture 2 of 11 (Chap 5, Matrices)

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OBJECTIVES

At the end of the lesson, students shouldbe able to:

(c) perform operations on matrices such

as multiplication of two matrices

(d) define the transpose of a matrix and

explain its properties

(b) define symmetric matrix and skew

symmetric matrix


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Multiplication of Matrices

The product of two matrices A and B is

defined only when the number of

columns in Ais equal to the number ofrows in B.

If order of A is mx nand the order of Bis nxp, then ABhas order mx p.


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The order of the

product is mx p

These numbers must

be equal

m

n np

AmxnBnxp ABmxp


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A row and a column musthave thesame number of

entriesin order to be

multiplied.

ATTENTION !


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Multiplication Of Two Matrices

naaaaR

321

nb

b

bb

C

3

2

1

1 1 2 2 3 3[ ]

n nRC a b a b a b a b


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Find :

12

43

12

502

321

Example 1


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12

43

12

502

321

)1(5)4(0)1(2)2(5)3(0)2(2

)1(3)4(2)1(1)2(3)3(2)2(1

36

122

Solution


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Find AB.

2 5 4

1 7 5

A

1 2 3 5

3 2 1 5

5 4 0 7

B

Example 2


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7 30 11 4345 4 10 5

AB

Solution


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Let and

Show that AB BA .

43

21A

23

12B

Example 3


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43

21

23

12

1 2 2 3 1 1 2 2

3 2 4 3 3 1 4 2

4 5

18 5

AB =

Solution


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BA =

=

=

2312

4321

)4(2)2(3)3(2)1(34)1()2(23)1()1(2

143

05


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Thus, AB BA .

This result prove that the matrixmultiplication is not commutative.

51854

14305


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Properties of Matrix Multiplication

Let A,B,C and D be matrices for which

the following products are defined .Then

ASSOCIATIVE PROPERTY

A(BC) = (AB)C DISTRIBUTIVE PROPERTY

A(B+C) = AB+AC


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Transpose Matrix

Definition

The transpose of a matrix A , written as AT

,is the matrix obtained by interchanging the

rows and columnsof A . That is, the

ith

column of AT

is theithrow ofA for all is.


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If Amxn= [aij] ,

then ATnxm= [aji]

11 12 13

21 22 23

31 32 33 3 3

a a a

A a a a

a a a

33332313

322212

312111

aaa

aaa

aaa

AT


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531

452

331

DIf then T =

543

353

121

133

1

2

BT

BLet then 31312

Example 4


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Let , and

Show that

(a) (A + B)T = A T+ BT

(b) (BC)T = CTBT

1 2

3 4A

3 4

2 1B

1 4

3 2C

Example 5


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(a) A + B =

1423

4231

55

64

(A + B)T =

56

54

=

Solution


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14

23

42

31

=

56

54

(A + B)T = AT+ BT

AT+ BT =


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) BC =

3 4 1 4

2 1 3 2

=

2822 =

25

[BC]T =

55


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Properties of transpose

(A B)T = AT BT(AT)T = A

(AB

)

T=

BTA

T

(kA)T = kAT ,kis a scalarTT

BA


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A square matrix,

A = [ aij]

nxn,

is symmetric if it is equal to its own

transpose,A = A T that is aij= aji

A Symmetric Matrix


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32

21,

32

21T

AA

A and B are symmetric matrices.

2

3

1

,

2

3

1

cbca

ba

Bcb

ca

ba

B

T

Example 6


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Given . Find AT.

Hence, prove that AATis a symmetric

matrix.

43

21A

Example 7


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42

31

43

21 T

TA

TAA

42

31

43

21

2511

115

Solution


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(AAT)T =

T

2511115

2511

115

= AAT

Since AAT= (AAT)T,

therefore AAT is a

symmetrical matrix


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A Skew Symmetric Matrix

A square matrix,

A = [ aij]nxn,

is a skew symmetric matrix if A = -A T

jiij aa where i j and 0iia


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02

20,

02

20TAA

A and B are skew symmetric matrix.

031

302

120

,

031

302

120

TBB

02

20A

B

Example 8


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A Symmetrical Matrix A = AT that is aij = aji

A = -AT, jiij aa A Skewed Symmetrical Matrix

Transpose Matrix

Properties of transpose

(A B)T = AT BT(AT)T = A(AB)T = BTAT(kA)T = kAT ,kis a scalar

TTBA


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Multiplication Of Two Matrices

naaaaR

321

nb

b

bb

C

3

2

1

1 1 2 2 3 3[ ]

n nRC a b a b a b a b


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032

421A

43

21B

462154

973

C

1. Let

,

and

Indicate whether the given product is defined.

If so, give the order of the matrix product.Compute the product, if possible.

(a) AB (b) AC (c) BA

(d) BC (e) CA (f) CB

Exercises:


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12

04

13

A

11

21

12

B

22

43C

2. Let

,

and

Find , (a) ATB (b) BTA

(c) (BC)T

(d) (A+B)T

A


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1.(a) Not defined

(b) Defined ; 2 x 3

(c) Defined ; 2 x 3

(d) Not defined.

(e) Not defined.

f Not defined

212918

274119

121811

485

Answers:


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01

1312

013

112

6810578

220

355

2 (a) (b)

(c) (d)

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Lecture 2 of 11 (Chap 5, Matrices)