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BA201 Engineering Mathematic
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B3001/UNIT7/1
____________________________________________________________________________________
Prepared by : Nur HIdayah Othman Page 1
Unit
7
MATRICES OPERATION
To know the different types of matrices and
understand how to apply it on simple algebra
problem solving.
Upon completing this module, you should be
able to:
1. Identify and determine the
determinant from the square matrix.
2. Identify and determine the minor
matrix from the square matrix.
3. Identify and determine the cofactor
matrix from the square matrix.
4. Identify and determine the inverse
matrix from the square matrix.
General Objectives
Specific Objectives
B3001/UNIT7/2
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7.0 INTRODUCTION
There are a few methods to solve simultaneous equations using the matrices such as Cramer’s
Rule and Inverse Matrices. Before we go further to look at the Cramer’s Rule and Inverse Matrices, we
should know and understand the operations of the matrices like determinant of the matrix, minor
matrices, co-factor, adjoint and inverse of square matrices.
7.1 DETERMINANT OF THE MATRIX
Determinant is a unique number that can be determined from the square matrix. It is used to
represent the real-value of the matrix which can be used to solve simple algebra problems later
on.
The symbol for the determinant of matrix A is det(A) or A.
7.1.1 Determinant of the matrix 2 x 2
For a matrix of size 2 x 2, the method to find the determinant is:
Let’s say, A =
dc
ba
Then, det(A) = A = dc
ba = (ad – bc)
Example 7.1:
If A =
87
65 determine det(A).
INPUT
B3001/UNIT7/3
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Prepared by : Nur HIdayah Othman Page 3
Solution:
A = (58 - 67)
= 40 – 42
= -2
Example 7.2:
If A =
95
24 determine det(A).
Solution:
A = (4·9 - 2·5)
= 32
= 12
7.1.2 Determinant of The Matrix 3 x 3
For a matrix of size 3 x 3, the method to find the determinant is:
If A =
333231
232221
131211
aaa
aaa
aaa
Then, A= 3231
2221
31
3313
2312
21
3332
2322
11aa
aaa
aa
aaa
aa
aaa
A= 312232213113233312213223332211 aaaaaaaaaaaaaaa
B3001/UNIT7/4
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Example 7.3 :
Determine of the determinant of matrix
212
034
231
Solution:
Determinant = 12
342
22
043
21
031
= 1(3·2 – 0·1) – 3(4·2 – 0·2) + 2(4·1 – 3·2)
= 6 – 24 – 4
= –22
Example 7.4 :
If A =
864
297
531
, determine A.
Solution:
A = 64
975
84
273
86
291
= 1(9·8 – 2·6) – 3(7·8 – 2·4) + 5(7·6 – 9·4)
= 60 – 144 + 30
= –54
B3001/UNIT7/5
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Prepared by : Nur HIdayah Othman Page 5
7a.1 Determine the determinant for the following matrices.
a.
124
136
b.
127
401
351
c.
35
83
7a.2 If A =
724
432
612
B =
36
24 dan C =
052
640
241
Determine:
i) A
ii) B
iii) C
ACTIVITY 7a
B3001/UNIT7/6
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Prepared by : Nur HIdayah Othman Page 6
FEEDBACK 7a
7a.1 a. 20
b. 1
c. -49
7a.2 i). 4
ii). 24
iii). -46
B3001/UNIT7/7
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Prepared by : Nur HIdayah Othman Page 7
7.2 MATRIX MINOR
The minor of a matrix is a new matrix where all the elements are determinants. Each
determinant is calculated by removing a row and a column from the original matrix. For
example, in order to determine the element at position ij, you will have to remove row i and
column j from the matrix. Next, you calculate the determinant of what is left.
If, A =
333231
232221
131211
aaa
aaa
aaa
then, Minor A =
333231
232221
131211
MMM
MMM
MMM
Where,
3332
2322
11aa
aaM by removing row 1 and column 1 from A
3331
2321
12aa
aaM by removing row 1 and column 2 from A
3231
1211
23aa
aaM by removing row 2 and column 3 from A
INPUT
B3001/UNIT7/8
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Example 7.5:
If A =
864
297
531
, determine minor A.
Solution:
The elements are:
86
2911 M = 9·8 – 2·6 = 60
84
2712 M = 7·8 – 2·4 = 48
64
9713 M = 7·6 – 9·4 = 6
86
5321 M = 3·8 – 5·6 = -6
84
5122 M = 1·8 – 5·4 = -12
64
3123 M = 1·6 – 3·4 = -6
29
5331 M = 3·2 – 5·9 = -30
27
5132 M = 1·2 – 5·7 = -33
97
3133 M = 1·9 – 3·7 = -12
Therefore, Minor A =
123330
6126
64860
B3001/UNIT7/9
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Example 7.6:
If P =
212
034
231
, determine minor P.
Solution:
The elements are:
21
0311 M = 3·2 – 0·1 = 6
22
0412 M = 4·2 – 0·2 = 6
12
3413 M = 4·1 – 3·2 = -2
21
2321 M = 3·2 – 2·1 = 4
22
2122 M = 1·2 – 2·2 = -2
12
3123 M = 1·1 – 3·2 = -5
03
2331 M = 3·0 – 2·3 = -6
04
2132 M = 1·0 – 2·4 = -8
34
3133 M = 1·3 – 3·4 = -9
Therefore, Minor P =
986
524
266
B3001/UNIT7/10
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Prepared by : Nur HIdayah Othman Page 10
7b.1 Determine the Minor of the following matrices:
i) A =
724
432
612
ii) B =
127
401
351
iii) C =
052
640
241
ACTIVITY 7b
B3001/UNIT7/11
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Prepared by : Nur HIdayah Othman Page 11
FEEDBACK 6b
7b.1 i)
82014
0105
163013
ii)
5120
372011
2278
iii)
4616
13410
81230
B3001/UNIT7/12
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7.3 CO-FACTOR OF A MATRIX
Once you have found the Minor of a matrix, you can easily determine the Cofactor of the matrix.
All the hard work is already done when you determine the Minor of a matrix. All you need to do
now is to multiply each element of the Minor of the matrix with a factor ji1 and the Cofactor
is done.
Let’s look at the following example:
If A =
333231
232221
131211
aaa
aaa
aaa
and Minor A =
333231
232221
131211
MMM
MMM
MMM
Therefore, Co-factor of a matrix A =
333231
232221
131211
KKK
KKK
KKK
Where, ij
ji
ij MK )1(
Then,
Co-factor of a matrix A =
33
6
32
5
31
4
23
5
22
4
21
3
13
4
12
3
11
2
111
111
111
MMM
MMM
MMM
INPUT
B3001/UNIT7/13
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Prepared by : Nur HIdayah Othman Page 13
Example 7.7:
If A =
864
297
531
, determine the co-factor of a matrix A
Solution:
First, find the minor of a matrix A,
123330
6126
64860
Next, multiply each element by its factor ji1
Therefore, the co-factor of a matrix A,
A =
121331301
6112161
61481601
654
543
432
A =
123330
6126
64860
B3001/UNIT7/14
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Example 7.8:
If P =
212
034
231
, determine the co-factor of a matrix P.
Solution:
First, find the minor of a matrix P,
986
524
266
Next, multiply each element by its factor ji1
Therefore, the co-factor of a matrix,
P =
918161
512141
216161
654
543
432
P =
986
524
266
B3001/UNIT7/15
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Prepared by : Nur HIdayah Othman Page 15
7c.1 Find the cofactor for the following matrices:
i) A =
724
432
612
ii) B =
127
401
351
iii) C =
052
640
241
AKTIVITY 6c
B3001/UNIT7/16
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Prepared by : Nur HIdayah Othman Page 16
FEEDBACK 7c
7c.1 i)
42014
0105
163013
ii)
5120
37201
2278
iii)
4632
13410
81230
B3001/UNIT7/17
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Prepared by : Nur HIdayah Othman Page 17
7.4 ADJOINT MATRIX
For a square matrix A with n x n, you can find the adjoint of a matrix when transposing the
cofactors of a matrix A.
In this case, for matrix A,
Adjoint of a matrix A, written as Adj(A) = KT where K is the co-factor for A
Then, if A =
333231
232221
131211
aaa
aaa
aaa
and Minor A =
333231
232221
131211
MMM
MMM
MMM
And co-factor matrix A =
333231
232221
131211
KKK
KKK
KKK
Then adjoint matrix A, Adj(A) =
332313
322212
132111
KKK
KKK
KKK
Or,
TKAAdj )(
INPUT
B3001/UNIT7/18
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Example 7.9:
If A =
864
297
531
, determine the ad joint of the matrix A.
Solution:
Inputs from example 4.14, you will find minor for A =
123339
6126
64860
And cofactor of A =
123339
6126
64860
Then, adjoint of matrix A, Adj(A) =
1266
331248
39660
B3001/UNIT7/19
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Example 7.10:
If P =
212
034
231
, determine the ad joint for matrix P.
Solution:
Inputs from example 4.15, the minor of P =
976
524
286
and co-factor of P =
976
524
286
with the adjoint for matrix P; Adj(P) =
952
728
646
B3001/UNIT7/20
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7D.1 Determine the adjoint for the following matrices:
i) A =
724
432
612
ii) B =
127
401
351
iii) C =
052
640
241
ACTIVITY 7d
B3001/UNIT7/21
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FEEDBACK 7d
7d.1 i)
4016
201030
14513
ii)
5372
12027
2018
iii)
4138
6412
321030
B3001/UNIT7/22
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7.5 INVERSE OF A SQUARE MATRIX
The inverse of a square matrix is its complement because when you multiply a matrix with its
inverse, the product is an Identify matrix.
In other words, if A is a square matrix and A-1
its inverse, then
AA-1
= I , where I is an Identity matrix.
In order to find A-1
, you can make use of the Adj(A) and A. The formula to find A-1
is
A-1
= A
1Adj(A)
Example 7.11:
If A =
864
297
531
, find A-1
.
Solution:
The determinant of A,
A = 64
975
84
273
86
291
= 60 – 144 + 30
= –54
INPUT
B3001/UNIT7/23
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The adjoint of A,
Adj(A) =
1266
331248
39660
Therefore, A-1
= A
1Adj(A)
= 54
1
1266
331248
39660
A-1
=
54
12
54
6
54
654
33
54
12
54
4854
39
54
6
54
60
Example 7.12:
If P =
212
034
231
, find P-1
.
Solution:
The determinant of P,
P = 12
342
22
043
21
031
= 1(3·2 – 0·1) – 3(4·2 – 0·2) + 2(4·1 – 3·2)
= 6 – 24 – 4
= –22
B3001/UNIT7/24
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The adjoint for matrix P,
Adj(P) =
952
728
646
Therefore, P-1
= P
1Adj(P)
= 22
1
952
728
646
P-1
=
22
9
22
5
22
222
7
22
2
22
822
6
22
4
22
6
B3001/UNIT7/25
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7e.1 Find the inverse of the following matrices:
i) A =
724
432
612
ii) B =
127
401
351
iii) C =
052
640
241
ACTIVITY 7e
B3001/UNIT7/26
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FEEDBACK 7e
7e.1 i)
40
4
40
0
40
1640
20
40
10
40
3040
14
40
5
40
13
ii)
121
5
121
37
121
2121
1
121
20
121
27121
20
121
1
121
8
iii)
142
4
142
13
142
8142
6
142
4
142
2142
32
142
10
142
30
B3001/UNIT7/27
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SELF ASSESSMENT
7.1 If A =
071
526
341
determine A .
7.2 If A =
071
526
341
determine minor for A.
7.3 If A =
071
526
341
determine cofactor for matrix A.
7.4 If A =
071
526
341
determine adjoint for matrix A.
7.5 If A =
071
526
341
determine the inverse for matrix A.