جامعة اإلمام محمد بن سعود اإلسالمية

كلية االقتصاد والعلوم اإلدارية

قسم التمويل واالستثمار

Al-Imam Muhammad Ibn Saud Islamic University College of Economics and Administration Sciences

Department of Finance and Investment

Financial Mathematics Course

FIN 118 Unit course

5 Number Unit

Matrices Unit Subject

Dr. Lotfi Ben Jedidia Dr. Imed Medhioub

we will see in this unit

Matrix / Matrices

Different types of matrices

Usual operations on matrices

2

Learning Outcomes

3

At the end of this chapter, you should be able to:

1. Define and use the matrix form.

2.Distinct between the different types of

matrices and its properties.

3. Make usual operations on matrices.

In Mathematics, matrices are used to store information. • This information is written in a rectangular arrangement of rows and columns. • Each entry, or element, of a matrix has a precise position and meaning.

Example1:

Matrix / Matrices

4

7 115

1204

0 312

4,3A

4 Columns

3

Ro

ws

Definition:

5

Example of Matrix • Food shopping online: people go online to shop four items

and have them delivered to their homes.

• Cartons of eggs, bread, bags of rice, packets of chickens were ordered online and the people left their address for delivery.

• A selection of orders may look like this:

Order

Address

Carton of

eggs

Bread Rice Chicken

Al Wuroud 2 1 3 0

Al Falah 4 0 2 1

Al Izdihar 5 1 1 7

6

Matrix notation A matrix is defined by its order which is always number of rows by number of columns.

• A horizontal set of elements is called a row • A vertical set is called a column • First subscript refers to the row number • Second subscript refers to column number

mnmmm

n

n

nm

aaaa

aaaa

aaaa

A

...

....

...

...

321

2232221

1131211

,

7

Matrix notation

• It has the dimensions m by n (m x n)

mnmmm

n

n

nm

aaaa

aaaa

aaaa

A

...

....

...

...

321

2232221

1131211

,

row 2

column 3

7 115

1204

0 312

4,3A

7 ,1 ,1 ,5

1 ,2 ,0 ,4

0 ,3 ,1 ,2

34333231

24232221

14131211

aaaa

aaaa

aaaa

8

Types of matrices

1/ Row vector m=1 2/ Column vector n=1

3/ Square Matrix m=n

nn rrrR .......21,1

m

m

c

c

c

C

.

.

2

1

1

7

3

1

1,3C

14234,1 R

nnnnn

n

n

nn

aaaa

aaaa

aaaa

A

...

....

...

...

321

2232221

1131211

,

721

54 3

32 1

3,3A

Some other Matrices

• Zero Matrix

• Transpose of a matrix

• Symmetric matrix

• Diagonal matrix

• Identity matrix

• Upper triangular matrix

• Lower triangular matrix

9

Zero Matrix

Every element of a matrix is zero, it is called a zero matrix, i.e.,

• The symbol O is used to denote the zero matrix.

10

0...00

......

0.....

0...00

O

11

Transpose of a matrix

The matrix obtained by interchanging the rows and columns of a matrix A is called the transpose of A (write AT or A’).

e.g.

mnmm

n

n

nm

aaa

aaa

aaa

A

...

......

......

......

...

...

21

22221

11211

,

nmnn

m

m

T

aaa

aaa

aaa

AA

...

......

......

......

...

...

21

22212

12111

'

724

3013,2A

73

20

41'

2,32,3 AAT

12

Symmetric matrix

• A symmetric matrix is a square matrix which

satisfy the following property:

• A symmetric matrix satisfy that : A = AT

Example: the following matrix is symmetric

a12 = a21 , a13 = a31

a23 = a32

.' ' sjandsiallforaa jiij

943

462

325

AAAT

943

462

325

13

Diagonal matrix

• A Diagonal matrix is a square matrix where all

elements off the main diagonal are zero.

Example:

jiwithsjandsiallforaa jiij ' ' 0

1000

0400

0030

0002

D

nn

nn

a

a

a

D

...000

....

0...00

0...00

22

11

,

14

Identity matrix

• An identity matrix is a diagonal matrix where all

elements of the main diagonal are one.

Examples:

jiallforaandia ijii 0 1

1000

0100

0010

0001

4I

1...000

....

0...010

0...001

nI

100

010

001

3I

10

012I

15

Upper triangular matrix

• An upper triangular matrix is a square matrix

where all elements below the main diagonal are

zero.

Examples:

jiallforaij 0

9000

6800

4120

1231

4U

300

710

542

3U

nn

nn

n

a

a

a

aaa

U

00

0

,1

22

11211

16

Lower triangular matrix

• A Lower triangular matrix is a square matrix

where all elements above the main diagonal are

zero.

Examples:

0 jiallforaij

9641

0812

0023

0001

4L

375

014

002

3L

nnnnn aaa

aa

a

L

1,1

2221

11

0

00

Usual operations on matrices

Addition and Subtraction of matrices

Matrices can be added or subtracted if they have the same order. Corresponding entries are added (or subtracted).

17

mnmnmn CBA

Examples

Consider

Find, if possible,

(1) A + B (2) A – C (3) B - A

(4) A + D (5) D – C (6) D - A

(7) C + D (8) C – D (9) D - B 18

43

21A

52

13B

54

03

21

C

11

12

10

D

19

Scalar Multiplication of matrices:

Multiplication of a matrix A by a scalar is obtained by multiplying every element of A by :

A = [ aij].

Examples:

mnmm

n

n

aaa

aaa

aaa

A

...

......

......

......

...

...

21

22221

11211

51

31

21

A

153

93

63

3A

255

155

105

5A

Usual operations on matrices

20

Usual operations on matrices

Multiplication of two matrices:

Multiplication of a matrix A by a matrix B is possible if the number of columns of A is equal to the number of rows of B.

Example:

A m x n B n x p = C m x p

4,24,33,2 CBA

21

Examples Consider

1/

2/

3/

43

21A

52

13B

54

03

21

C

54132433

52112231

52

13

43

21BA

BAAB

43

21

52

13

52

13

54

03

21

AC

Example of use • Let’s show, by a real-life example, why matrix

multiplication is defined in such a way.

• Ahmad, Muhammad and Nayef three students in IMAM University invest their money in the same stocks but with different quantities. Ahmad’s, Muhammad's and Nayef’s stock holding are given by the matrix:

22

10040

102010

103020

321

Nayef

Muhammad

Ahmad

QQQ

M

Example of use (continued)

• Now suppose that the prices of three types of stocks are respectively 25 SAR, 22 SAR and 30 SAR.

• Let the price matrix be

• Then the amount that the first student will pay is just the first entry of the product.

23

30

22

25

P

1300

990

1460

30

22

25

10040

102010

103020

PM

Matrix Properties • Matrix addition is commutative : A + B = B + A

• Matrix addition is associative: A + (B +C) = (A + B) +C

• Matrix addition is distributive : α(A + B) = α A + α B

where α is a scalar.

• Matrix multiplication is associative : (A.B).C = A.(B.C)

• Matrix multiplication is distributive : (A + B).C = A.C + B.C

• Multiplication is generally not commutative : A.B ≠ B.A

24

Time to Review !

Matrices are used to store information. This information is written in a rectangular arrangement of rows and columns.

Matrices can be added or subtracted if they have the same order.

Multiplication of a matrix A by a matrix B is possible if the number of columns of A is equal to the number of rows of B.

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we will see in the next unit

Determinant of matrices

Some Properties of Determinants

Some applications of determinant of

matrices

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