C HAPTER 1 Matrix Algebra 1. I NTRODUCTION A rectangular array of elements or entries aij involving...
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Transcript of C HAPTER 1 Matrix Algebra 1. I NTRODUCTION A rectangular array of elements or entries aij involving...
1
CHAPTER 1Matrix Algebra
2
INTRODUCTION
• A rectangular array of elements or entries aij involving m rows and n columns
• Matrix can be written as A = [aij ]
Matrix
11 12 13 1 1
21 22 23 2 2
31 32 33 3 3
1 2 3
1 2 3
. .
. .
. .
. . . . . . . [ ]
. .
. . . . . . .
. .
j n
j n
j n
ij m n
i i i ij in
m m m mj mn
n coloumns
a a a a a
a a a a a
a a a a a
A m rows a
a a a a a
a a a a a
3
Elements in matrix may be real or complex numbers, or even functions.
1 3 2 real number
2 9 1
2 3 4complex number
1 2
i i
i
cos sin function
sin cos
4
If i = j, then the elements is called the leading diagonal of matrix A.
Example 11 22 33, ,a a a
5
EXAMPLE
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SOLUTION
i. Order of matrix : 3x3ii. Leading diagonal : 1,0,3iii. a12 = 2
a31= 5
a23= 2
7
TYPE OF MATRICES
8
Not diagonal
Not diagonal
Diagonal
9
Scalar
Scalar Not scalar
10Not identity matrix
Not identity matrix
11
2 4
2 7A
3 0
9 3
0 1
B
12
13
14
1 3 7
2 3 6
4 0 8
TA
9 1
3 2 4TB
15
T
5 1 3
1 2 2 since A =A, so it is a symmetric matrix
3 2 7
TA
16
17
T
0 1 5 0 1 5
1 0 3 , - 1 0 3
5 3 0 5 3 0
since A =-A, so it is a skew symmetric matrix
TA A
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Row echelon form
Not Row echelon form
Row echelon form
Not Row echelon form
19
Reduced Row echelon form
Reduced Row echelon form
Not Reduced Row echelon form
Not Reduced Row echelon form
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i) Not possible
1 2 3 7 8 9ii)
4 5 6 10 11 12
8 10 12
14 16 18
A B
A B
MATRIX OPERATION
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i) Not possible 1 5 0
)10 1 2
ii B A
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MATRIX OPERATION
23
24
2(3) 2(1) 2( 1)2
2(0) 2( 2) 2(4)
6 2 22
0 4 8
A
A
25
3(1) 1(2) 4( 6) 3( 1) 1(0) 4(4))
5(1) 2(2) ( 2)( 6) 5( 1) 2(0) ( 2)(4)
19 13
21 13
i AB
AB
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27
3 2 3
3 0 1AB
Answer
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DETERMINANT
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)det
3(1) 6( 1)
9
i A A
)det
3 1
6 1
3(1) ( 1)(6)
=9
T T
T
T
ii A A
A
A
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+++---
31
3( 5)(7) ( 1)(2)(0) 6(9)(4) 6( 5)(0) (2)(4)(3) ( 1)(9)(7)
111 39
150
A
A
A
32
33
11
211
11
2 2)
3 0
2(0) ( 2)( 3)
6
( 1) ( 6)
6
i M
C
C
34
35
11 11 12 12 13 13
2 3 45 2 9 2 9 5(3) ( 1) ( 1) ( 1) (6) ( 1)
4 7 0 7 0 4
129 63 216
150
A a c a c a c
A
A
A
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37
i) Find all the value of Cij
ii) Arrange all the value in matrix form [Cij]
iii) Then transpose [Cij]
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11 12 13
21 22 23
31 32 33
2, -6, 1
3, 0, 3
1, 3, 5
2 6 1 2 3 1
3 0 3 , then adj 6 0 3
1 3 5 1 3 5
T
c c c
c c c
c c c
Cij Cij
39
40
1
) 3(6) 4(2) 10
6 41
2 310
i A
A
41
42
11 12 13
21 22 23
31 32 33
1
0, 9, 18
2, 6, 11
1, 3, 1
0 9 18 0 2 1
2 6 11 , then adj 9 6 3
1 3 1 18 11 1
0 2 11
9 6 39
18 11 1
T
c c c
c c c
c c c
Cij Cij
A
43
44
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EXERCISE:
Find Inverse of matrix A using elementary row operation
1 0 1
1 2 3
3 1 5
A
1
7 11
2 22 1 1
5 11
2 2
A
Answer
46
4 4 2
0 5 1
4 3 2
B
Find Inverse of matrix B using elementary row operation
1
7 1 3
4 2 21 0 1
5 1 5
B
Answer
47
SOLVING THE SYSTEMS OF LINEAR EQUATION
1.6.1 Inversion Method
1.6.2 Gaussian Elimination and Gauss-Jordan Elimination
1.6.3 Cramer’s Rule
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1.6.1 INVERSION METHOD
AX = B
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1 1A AX A B
1 1( )
A AX A A X
X
1X A B
The left-hand side can be simplified by noting that multiplying a matrix by its inverse gives the identity matrix
Multiplying a matrix by the identity matrix has no effect and so
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1
7 2 12
3 1 5
1 21
3 77(1) 2(3)
1 2
3 7
AX B
x
y
A
1
,
1 2 12
3 7 5
1(12) ( 2)(5)
( 3)(12) 7(5)
2
1
, 2, 1
Then
X A B
x
y
hence x y
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1 2 3
1 2 3
1 2 3
3 6
8 9 4 21
2 2 3
x x x
x x x
x x x
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1.6.2 GAUSSIAN ELIMINATION AND GAUSS-JORDAN ELIMINATION
Gaussian elimination process.• The procedure to reduce matrix A to
row echelon form (REF) by using elementary row operation (ERO).
Gauss-Jordan elimination• The procedure to reduce matrix A to
reduced row echelon form (RREF) by using elementary row operation (ERO).
54
EXAMPLE 1.6.2i. Solve the system of linear equations by using
Gaussian elimination and Gauss-Jordan elimination.
1 2 3
1 2 3
1 2 3
2 2 8
3 3 4
4 2 1
x x x
x x x
x x x
55
GAUSSIAN ELIMINATION
2 1
3 1
23 2
3
22 1 4
1147
1
5
2 1 2 8 1 3 3 4
1 3 3 4 2 1 2 8
4 2 1 1 4 2 1 1
1 3 3 41 3 3 4
4 160 7 4 16 0 1
7 70 14 13 17
0 14 13 17
1 3 3 4 1 3
4 160 1
7 70 0 5 15
R RR R
R R R
R
R R
3 4
4 160 1
7 70 0 1 3
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1 2 3
2 3
3
2
2
1
1
From the Gaussian elimintion we have
3 3 4
4 16
7 7 3
4 16(3)
7 74
3(4) 3(3) 4
1
x x x
x x
x
x
x
x
x
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2 1
3 1
23 2
22 1 4
1147
From Gauss-Jordan Elimination,
2 1 2 8 1 3 3 4
1 3 3 4 2 1 2 8
4 2 1 1 4 2 1 1
1 3 3 41 3 3 4
4 160 7 4 16 0 1
7 70 14 13 17
0 14 13 17
1 3 3 4
4 160 1
7 70
R RR R
R R R
R R
31 2
135
1 3 3 4
4 160 1
7 70 5 15 0 0 1 3
R R R
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1 3
2 3
9
74
7
1 2 3
9 201 0
7 7 1 0 0 14 16
0 1 0 1 0 47 7
0 0 1 30 0 1 3
1, 4, 3
R R
R R
x x x
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60
1.6.3 CRAMER’S RULE
method of obtaining the solution of equations like these as the ratio of two determinants.
11211 byaxa 22221 byaxa
1 12
2 22
11 12
21 22
i
b aA
b ax
a aA
a a
11 1
12 2
11 12
21 22
i
a bA
a by
a aA
a a
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62
63
1 3 1 8
2 1 1 7
1 1 1 2
,
8 3 1
7 1 1
2 1 1 162
1 3 1 8
2 1 1
1 1 1
x
y
z
then
x
1 8 1
2 7 1
1 2 1 12 31 3 1 8 2
2 1 1
1 1 1
y
1 3 8
2 1 7
1 1 2 12 31 3 1 8 2
2 1 1
1 1 1
z
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1.7 MATRIX APPLICATION
1.7.1 Application of Inversion Method
1.7.2 Application of Gaussian Elimination and Gauss Jordan Elimination Method
1.7.3 Application of Cramer’s Rule
65
EXAMPLE 1.7.1.2 - APPLICATION OF INVERSION METHOD Grand Canyon Tours offers air and ground scenic tours of
the Grand Canyon. Tickets for the 7.5 hours tour cost RM169 for an adult and RM 129 for a child and each tour group is limited to 19 people. On three recent fully booked tours, total receipts were RM2931 for the first tour, RM3011 for the second tour and RM 2771 for the third tour. Determine how many adults and how many children were in each tour.
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Let A= Adult
C= Children
From information above, the equation are
1
19
169 129 3011
Tour
A C
A C
1 1 19
169 129 3011
129 1 191
169 1 30111 129 1 169
129 1 191
169 1 301140
129 11940 40
169 1 3011
40 40
A
C
A
C
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14
5
14,
5
A
C
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EXAMPLE 1.7.2.1 – APPLICATION OF GAUSSIAN ELIMINATION AND GAUSS JORDAN ELIMINATION METHOD
Ahmad inherited RM25000 and invested part of it in a money market account, part in municipal bonds and part in a mutual fund. After one year, he received a total of RM1620 in simple interest from three investments. The money market paid 6% annually, the bonds paid 7% annually and the mutually fund paid 8% annually. There was RM6000 more invested in the bonds than the mutual funds. Find the amount Ahmad invested in each category using Gauss Elimination Method.
69
SolutionLet A= Money Market Account
B=Municipal bond C=Mutual fund
Arrange in matrix form
25000
0.06 0.07 0.08 1620
6000
A B C
A B C
B C
1 1 1 25000
0.06 0.07 0.08 1620
0 1 1 6000
70
2 1 2
3 23
1 1 1 25000 1 1 1 25000
0.06 0.07 0.08 1620 0.06 0 0.01 0.02 120 100
0 1 1 6000 0 1 1 6000
1 1 1 25000 1 1 1 25000
0 1 2 12000 0 1 2 12000
0 1 1 6000 0 0 3 6000 1
3
1 1 1
0 1 2
R R R
R R R
HHHHHHHHHHHHHH
HHHHHHHHHHHHHH
25000
12000
0 0 1 2000
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25000
25000 8000 2000
15000
A B C
25000
2 12000
2000
A B C
B C
C
12000 2
12000 2(2000)
12000 4000
8000
B C
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EXAMPLE 1.7.3.1-APPLICATION OF CRAMER’S RULE
A salesman has the following record of sales during three months for three items A, B and C which have different rates of commission. Find out the rates of commission on the items A, B and C. Solve by Cramer’s rule.
Months Sales of Units Total commission drawnA B C
JanuaryFebruaryMarch
90 100 20 130 50 4060 100 30
800900850
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SolutionChange into matrix form
800 100 20
900 50 40
850 100 30
90 100 20
130 50 40
60 100 30
50 40 900 40 900 50800 100 20
100 30 850 30 850 100
50 40 130 40 130 5090 100 20
100 30 60 30 60 100
A
800 2500 100 7000 20 47500
90 2500 100 1500 20 10000
350000
175000
2
90 100 20 800
130 50 40 900
60 100 30 850
A
B
C
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90 800 20
130 900 40
60 850 30
90 100 20
130 50 40
60 100 30
900 40 130 40 130 90090 800 20
850 30 60 30 60 850
175000
90 7000 800 1500 20 56500
175000
700000
175000
4
B
75
90 100 800
130 50 900
60 100 850
90 100 20
130 50 40
60 100 30
50 900 130 900 130 5090 100 800
100 850 60 850 60 100
175000
90 47500 100 56500 800 10000
175000
1925000
175000
11
C
76
1.8 INPUT OUTPUT MODEL
agriculture (A), manufacturing (M),and services (S)
Column A. Production of 1 unit agricultural products
requires the consumption of 0.2 unit of agricultural
products, 0.2 unit of manufactured goods and 0.1 units of
services.
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Example 1.8.1TKK Corporation, a large conglomerate, has three subsidiaries engaged in producing raw rubber, manufacturing tires and manufacturing other rubber based goods. The production of 1 unit of raw rubber requires the consumption of 0.08 unit of rubber, 0.04 units of tires and 0.02 unit of other rubber based goods. To produce 1 unit of tires requires 0.6 unit of raw rubber, 0.02 unit of tires and 0 units of other rubber based goods. To produce 1 unit of other rubber based goods requires 0.3 unit of raw rubber, 0.01 units of tires and 0.06 unit of other rubber based goods. Markets research indicates that the demand for the following year will be RM 200 million for raw rubber, RM 800 million for tires and RM 120 million for other rubber based products. Find the level of production for each subsidiary in order to satisfy this demand.
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Solution
80
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