fin118 unit 6

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

6 Number Unit

Applications of Matrices Unit Subject

Dr. Lotfi Ben Jedidia Dr. Imed Medhioub

we will see in this unit

Determinant of matrices

Some Properties of Determinants

Some applications of determinant of

matrices

2

Learning Outcomes

3

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

1. Identify square matrices and its regularity.

2. Find the determinant and the inverse of

matrices

3.Solve the system of linear equations by Cramer’s

Rule.

In Mathematics, square matrices play prominent role in the application of matrix algebra to real-world problems. • A square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order n. Any two square matrices of the same order can be added and multiplied.

Example1:

Square Matrices

4

115

204

312

3,3A

3 Columns

3

Ro

ws

Definition:

5

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

three items and have them delivered to their homes.

• Cartons of eggs, bread, bags of rice 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

Al Wuroud 2 1 3

Al Falah 4 0 2

Al Izdihar 5 1 1

6

Remember a Square Matrix notation

A square 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

nnnnn

n

n

nn

aaaa

aaaa

aaaa

A

...

....

...

...

321

2232221

1131211

,

7


The determinant of square matrix A, denoted det(A),

or , is a number that is evaluated by all elements of

A.

Determinant of order 2

The determinant of a 2x2 matrix is the difference between the product of the major diagonal elements and the product of the minor diagonal elements .

A

12212211

2221

1211det aaaa

aa

aaA

Definition 1:

8

More Examples

Evaluate the determinant of each matrix:

1/

2/

3/

4/

42

21A

42

42B

76

51C

52

12D

9

Determinant of order 3:

• To find the determinant of a 3 x 3 matrix, first recopy the first two columns. Then we obtain 3 major diagonal elements and 3 minor diagonal elements (Rule of Sarrus).

• The determinant of a 3x3 matrix is the difference between the sum of the products of the major diagonal elements and the sum of the products of the minor diagonal elements .

122133112332132231322113312312332211

32

22

12

31

21

11

333231

232221

131211

det aaaaaaaaaaaaaaaaaa

a

a

a

a

a

a

aaa

aaa

aaa

A


Find the determinant of the following matrix

10

50003000080000120004000

0

20

30

40

10

20

10040

102010

103020

det M

10040

102010

103020

M

Example:


Evaluate the determinant of theses matrices:

11

01 01300

1020 990

10301460

1 P

01130040

10 990 10

10146020

2 P

1300 040

990 2010

14603020

3 P

Examples: Determinant of matrices

12

Determinant of order 3:

An alternative method involves finding the determinant in terms of three 2x2 matrices. This alternative method will work for all sizes, not just a special case for a 3×3 matrix (Expansion by Minors ).

132223123113323312212332332211

122133112332132231322113312312332211

333231

232221

131211

det

aaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaaa

aaa

aaa

aaa

A

333231

232221

131211

31

333231

232221

131211

21

333231

232221

131211

11

333231

232221

131211

det

aaa

aaa

aaa

a

aaa

aaa

aaa

a

aaa

aaa

aaa

a

aaa

aaa

aaa

A


13

Evaluate the determinant of theses matrices with

expansion by minors:

01 01300

1020 990

10301460

1 P

01130040

10 990 10

10146020

2 P

1300 040

990 2010

14603020

3 P

Examples: Determinant of matrices

18

Some Properties of Determinants

P1: det (A) = det (AT)

P2: If all entries of any row or column is zero, then

det (A) = 0

P3: If two rows or two columns are identical, or

linearly dependent then det (A) = 0

P4: If A is a diagonal matrix or upper triangular or

lower triangular matrix, then det(A) is equal to

the product of all diagonal elements.

P5: if , then A is regular and invertible

ii

n

iaA

1

0A

26

Some applications of determinant of matrices

• Mr. Cramer tells us that we can use determinants to solve a linear system. (No elimination;No substitution!)

Cramer’s Rule on a System of Two Equations

Let A be the coefficient matrix for the system:

If det(A) 0, then the system has one solution, and

2222121

1212111

bxaxa

bxaxa

2221

1211

aa

aaA

A

ab

ab

x222

121

1 A

ba

ba

x221

111

2

Gabriel

Cramer

(1750ish)

S = {(x1 , x2)}

27

Example

Solve the system using Cramer’s Rule. Step1: The coefficient matrix for the system and its determinant are:

Step2:

S = {(-11, 9)}

1253

532

21

21

xx

xx

A

1x

A

2x

Cramer’s Rule on a System of Two Equations

28

Let A be the coefficient matrix for the system:

If det(A) 0, then the system has one solution, and

A

aab

aab

aab

x33323

23222

13121

1

333231

232221

131211

aaa

aaa

aaa

A

3333232131

2323222121

1313212111

bxaxaxa

bxaxaxa

bxaxaxa

A

aba

aba

aba

x33331

23221

13111

2 A

baa

baa

baa

x33231

22221

11211

3

Cramer’s Rule on a System of Three Equations

S = {(x1 , x2, x3)}

29

Example:

Solve the system using Cramer’s Rule.

Step1: The coefficient matrix for the system and its determinant are:

130 4

992

14632

31

321

321

xx

xxx

xxx

A A


30

Step2:

1x 2x

3x


S = {( 25, 22, 30)}

Time to Review !

Matrices are used to transcript information in a system of equations

The determinants of Matrices can be used to solve a linear system. (No elimination; No substitution ! Cramer rule)

36

we will see in the next unit

The “arithmetic sequences” and

“arithmetic series”.

The “Geometric sequences” and

“Geometric series”.

Solve some questions for real world

situations in order to solve problems,

especially economic and financial.

21

fin118 unit 6

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