Mathematics for Management Chapter 2: Matrix Algebra

Mathematics for Management by Nor Alisa Mohd Damanhuri http://ocw.ump.edu.my/course/view.php?id=440

Mathematics for Management

Chapter 2: Matrix Algebra

by

Nor Alisa Mohd Damanhuri and Ezrinda Mohd Zaihidee Faculty of Industrial Sciences & Technology

http://ocw.ump.edu.my/course/view.php?id=440


Content:

2.1 Matrix Notation and Terminologies

2.2 Types of Matrices

2.3 Matrix Operations

2.4 Reduced Matrix

2.5 Determinant

2.6 Inverse Matrix



Expected Outcome:

Upon the completion of this course, students will have the ability to:

1. Apply the knowledge to solve the identified matrix algebra problems such as manipulate matrix algebra and determinants, apply row operations and elementary matrices.



Matrix Notation & Terminology

Definition (m x n Matrix)

a rectangular array of numbers enclosed within brackets that consisting of m horizontal row and n vertical columns,

where ij denotes the entry in the ith row and jth column.

A matrix usually denoted by bold capital letters. Eg:A,B,C

mnmm

n

n

nmij

aaa

aaa

aaa

a

...

......

......

......

...

...

21

22221

11211

A



Type of Matrices

5 5





6





7





8



Transpose of a Matrix

The transpose of m x n matrix A, denoted by ,

is the n x m matrix whose ith row is the ith column of A.

interchange its rows with its column.

9 9 9

TA



Example:

10

Solution:



Exercises:

11



Equality of Matrices

Matrices and are equal if and only if

they have the same size and for each i and j

(corresponding entries are equal).

ijbB ijaA

12 12

ijij ba



Example:

13



Exercises:

14



Matrix Operations

1. Matrix Addition and Subtraction

If and are both m x n matrices, then

the is the m x n matrix obtained by adding or

subtracting corresponding entries of A and B, that is

15 15

ijaA

ijij ba BA

BA

15

ijbB



16

If A, B, C and O have the same size,

(a) (commutative) (b) (associative) (c) (identity)

ABBA

CBACBA

AAOOA

Matrix Properties



Example:

Compute the following

a)

b)

41

10

01

65

43

21

1

2

43

01

17 17



Solution:

18



Exercise:

19



If A is an m x n matrix and is a real number (also called a

scalar), then by kA, we denote the m x n matrix obtained

by multiplying each entry in A by k, that is

20 20 20

2 Scalar Multiplication



Properties

21



Example:

22

Solution:



Exercise:

23



Let A be an m x n matrix and B be an n x p matrix. Then

the product AB is the m x p matrix C whose entry in row

i and column j is obtained as follows: Sum the products

formed by multiplying, in order, each entry (that is, first, second,

etc.) in row i of A by the “corresponding” entry (that is, first,

second, etc.) in column j of B.

A B = C

m x n n x p m x p

24 24

3 Matrix Multiplication



Properties

i. A(BC) = (AB)C (associative)

ii. A(B + C) = AB + AC or (A+B)C = AC + BC (distributive)

25 25



Example:

26



Solution:

27



Exercise:

28



Reduced Matrix

A matrix is said to be a reduced matrix, provided that all of the following are true:

• If a row does not contain entirely of zeros, then the first nonzero entry in the row, called the leading entry, is 1, whereas all other entries in the column in which the 1 appears are zeros.

• The first nonzero entry in each row is to the right of the first nonzero entry in each row above it.

• Any rows that consist entirely of zeros are at the bottom of the matrix

29 29



Example:

For each of the following matrices, determine whether it is

reduced or not reduced.

1) 2) 3) 4)

5) 6) 7)

010

001

20

01

30 30

00

10

01

000

000

100

000

001

0000

3100

2010

01

10



Solution:

31



Exercise:

32



Elementary Row Operation

An augmented coefficient matrix is transformed into a row-equivalent

matrix if any of the following row operation is performed.

i) two rows are interchanged:

ii) a row is multiplied by a non zero constant:

iii) a constant multiple of one row is added to another row

jji RRkR

ji RR

33 33

ii RkR



34

ji RR

ii RkR

jji RRkR



Types of Solutions

1) If the reduced augmented coefficient matrix has a row of

the form

where k is a nonzero constant, then the linear system

AX = B has no solution and inconsistent.

k000

35



36

2) If the reduced augmented coefficient matrix has a last row

of the form

then the linear system AX = B has infinite number of

solutions.

3) If the reduced augmented coefficient matrix has a last row of

the form

then the linear system AX = B has unique solutions.

36

lk00

lkj0



Solving System by

Reduced Matrix

37



38



Example:

39



Solution:

40





41



42



Exercise:

43



Non-homogeneous and Homogeneous System

Definition:

The system

is called a homogeneous system if .

The system is a non-homogeneous system if at least one of

the c is not equal to zero.

mnmnmm

nn

nn

cxaxaxa

cxaxaxa

cxaxaxa

...

. . . .

. . . .

. . . .

...

...

2211

22222121

11212111

0...21 mccc

44 44



45



Exercise:

46



Determinant

2x2 Matrix

47 47



Example:

48

Solution:



Exercise:

49



3x3 Matrix

50



Example:

51



Exercise:

52



Inverse

53

• Objective: To determine the inverse of an invertible

matrix and to use inverses to solve system.

• Definition: If A is a square matrix and there exists a

matrix C such that CA=I, then C is called an inverse of

A, and A is said to be invertible.



2.6.1) Inverse by Adjoint Method

If A is a square matrix, the transpose matrix of matrix cofactor

A is known as adjoint of matrix A and is denoted by ,

where C is the cofactor of matrix A. If (non singular),

then the inverse matrix

TCA Adj

0A

54

AA

A Adj11



2x2 Matrix

In the case of a 2x2 matrix, a simple formula exists to find

its adjoint matrix

If then

55

dc

baA

ac

bdAAdj )(



56

Example:

Solution:





3x3 Matrix

• Consider the matrix A of order 3 x 3,

Suppose that we choose any entry, say , and strike out the row

and column that pass through , then we will obtain

which called the minor of the entry

•

333231

232221

131211

aaa

aaa

aaa

A

32133312

3332

1312

21 aaaaaa

aaM

57

21a

21a



58

So, we can get the adjoint matrix,



59

Example:



60

Solution:



61

Example:



62

Solution:



63





Exercises:

64



Inverse Matrix by Elementary Row Operation

If can be transformed by elementary row operation to ,

then the resulting matrix M is .

If a matrix A does not reduce to I, then does not exist.

IA -1AI

65

1A



Example:

66



Solution:

67





68





69



Exercise:

70



Solving System by Inverse Matrix

Solve the system

by finding the inverse of the coefficient matrix.

1102

2 24

1 2

321

321

31

xxx

xxx

xx

71



Solve by using adjoint matrix

72





73





Solve by using ERO

74



75





76



Exercise:

77



THE END ~THANK YOU~

78



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Mathematics for Management Chapter 2: Matrix Algebra