fin118 unit 6
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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
Determinant of matrices
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
Determinant of matrices
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:
Determinant of matrices
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
Determinant of matrices
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
Cramer’s Rule on a System of Three Equations
30
Step2:
1x 2x
3x
Cramer’s Rule on a System of Three Equations
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
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