topic 2 matrices & systems of linear equations
DESCRIPTION
Topic 2 MATRICES & systems of linear equations. LEARNING OUTCOMES. At the end of this topic, student should be able to : D efination of matrix - PowerPoint PPT PresentationTRANSCRIPT
TOPIC 2MATRICES &
SYSTEMS OF LINEAR EQUATIONS
LEARNING OUTCOMESAt the end of this topic, student should be able to : Defination of matrix Identify the different types of matrices such as rectangular, column,
row, square , zero / null , diagonal, scalar, upper triangular, lower triangular and identity matrices
Solve the equality of matrices Perform operations on matrices such as addition, subtraction, scalar
multiplication of two matrices Identify Transpose of Matrix Define the determinant of matrix and find the determinant of 2 x 2 and 3 x 3 matrix Find minor and cofactor of 2 x 2 and 3 x 3 matrix Define Inverse Matrix Find the Inverse Matrix by using Adjoint Matrix Write a system of linear equations Solve the system of linear equations by using Inverse Matrix and
Cramer’s Rule
Definition of Matrix
A matrix is an ordered rectangular array of numbers or functions.
The numbers or functions are called elements or the entries of the matrix.
Example of a matrix
A = Order of a matrix = row x column = (m x n)
Order of A matrix = 3 x 3Element of A matrix = 1,2,3,4,5,6,7,8,9
987654321
741
31
21
11
aaa
852
32
22
12
aaa
963
33
23
13
aaa
Types of Matrices
1. Rectangular Matrix A = 2 x
4
2. Column Matrix A = 2 x 1
87654321
21
3. Row Matrix A = 1 x 3
4. Square Matrix A = 2 x 2
5. Diagonal Matrix A = 3 x 3
321
3640
700050001
6. Zero / Null Matrix A= 2 x 2
7. Scalar Matrix A = 3 x 3
8. Upper Triangular Matrix A = 3 x 3
0000
200020002
600540321
9. Lower Triangular Matrix A = 3 x 3
10. Identity Matrix A = 3 x 3
OR A = 2 x 2
654032001
100010001
1001
Equality of Matrices
Two matrices are equal if they have the same order and same entries.
Exercises1. Find the value of x and y for the following :
(a)
(b)
(c)
(d)
4350
35y
x
10243
523yxyx
yx
3221
921
zy
yxx 4
322
Operations on MatricesAdditions / SubtractionsAdditions or subtractions of matrices can be done if they have the same dimensions whereby the two matrices must have the same number of rows and the same number of columns.
When two matrices are added or subtracted then the order of matrix should be the same.
MultiplicationScalar MultiplicationExample : A =
2A = =
4771
4771
2
814
142
Multiplication of Two MatricesNecessary condition for matrix multiplication Column of first matrix should be equal to the row of
the second of matrix.
Example :
0021
0432
0024001403220312
8442
Exercises1. A = B = C =
Find : (a) A + C (b) C – A (c) 3A – 2C (d) A + 2C (e) AB
201
043
431510216
123512
Transpose of a Matrix
A matrix obtain by interchanging the rows and and columns of the original matrix. It is denoted by or .
If is an is an matrix, that is the matrix.
TA 'A
A nm TAmn
Determinant of MatrixThe determinant of matrix is a unique real number for every square matrix.The determinant of a square matrix is denoted by Det A or .
Determinant of Matrix 2 x 2 Let us consider a 2 x 2 matrix :
2221
1211
aaaa
A
21122211 aaaaA
A
Example :Find the value of the determinant for matrix A.
Solution :
5973
A
)97()53( A
48
Determinant for Matrix 3 x 3Let us consider a 3 x 3 matrix :
333231
232221
131211
aaaaaaaaa
A
3231
222113
3331
232112
3332
232211 aa
aaa
aaaa
aaaaa
aA
Example :
Given , find or determinant of A.
Solution :
=
512214
321A A
1214
35224
)2(5121
1
A
)24(3)420(2)25(1
35
Minor of 2 x 2 Matrix
Let us consider matrix 2 x 2,
2221
1211
aaaa
A
2112
2211
aMaM
1234
4321
22
21
12
11
MMMM
A
ijij MM
Minor of 3 x 3 Matrix
Let us consider matrix 3 x 3,
333231
232221
131211
aaaaaaaaa
A
3332
232211 aa
aaM
32233322 aaaa
2321
131132 aa
aaM
21132311 aaaa
Example :
Find and .
Solution:
(a)
(b)
416342321
A
133164134
11 M
0444221
33 M
11M 33M
Cofactor
Cofactor for 2 x 2 Matrix
Let us consider matrix 2 x2
ijji
ij MC )1(
2221
1211
aaaa
A
....)1( 222
11 aC....)1( 21
312 aC
....)1( 123
21 aC
....)1( 114
22 aC
Example :
Given , find the cofactor for A.
Solution :
41028
A
4)4()1( 211 C
10)10()1( 312 C
2)2()1( 321 C
8)8()1( 422 C
82104
ijC
Cofactor for 3 x 3 Matrix
Let us consider matrix 3 x 3 ,
=
333231
232221
131211
aaaaaaaaa
A
3332
2322211 )1(
aaaa
C
32233322 aaaa
Example :
Given , find cofactor for A.
Solution :
3365382104
A
241593353
)1( 211
C
6)3024(3658
)1( 312
C
4218243638
)1( 413
C
924564802442624
ijC
Inverse Matrix by using Adjoint Matrix
where Steps :1. Find 2. Identify3. Identify 4. Substitute and Adj A in Inverse Matrix formulae.
AdjAA
A 11
tijCAdjA
A
ijC tijCA
Example 1 :Find if .
Solution: Step 1 :
Step 2 :
Step 3 :
Step 4 :
1A
4321
A
264 A
1234
ijC
1324t
ijC
1324
211A
21
23
12
Example 2 :Find if .
Solution: Step 1 :
Step 2 :
Step 3 :
Step 4 :
1B
125323214
B
2523
215
331
1232
4
B
34
11373145
16184
ijC
1131631418
754t
ijC
1131631418
754
3011A
3011
101
158
101
157
53
307
61
152
Systems of Linear EquationsA system of linear equations is a collection of two @ more linear equations, each containing one or more variables.
The following is a system of three equations containing three variables.
Using a matrix notation, we can write this system in simplified form.
This is called the augmented matrix of the system.
532216234
1
zyxzyx
zyx
516
1
322234111
Exercise
Write the augmented matrix of each system.
(a)
(b)
532643
yxyx
08201
02
yxzx
zyx
Solving a system using an Inverse MatrixConsider the pair of simultenous equations
Let the matrix of coefficient be , that is
In matrix form of system above can be written as
qdycxpbyax
A
dcba
A
BAX
BAX1
ExampleSolve the following equations by using Inverse Matrix.
Solution
Step 1: Step 5 : Step 2 :
Step 3 :Step 4 :
666723
yxyx
6
76623yx
A X B
30)12(18 A
BAX 1
3266
ijC
3626t
ijCAdjA
3626
3011A
67
101
51
151
51
yx
21
yx
21
yx
(b) Solving a system using Cramer’s RuleConsider the pair of simultenous equations
Let the matrix of coefficient be , that is
Therefore by using Cramer’s Rule
for 2 x 2 Matrix
Aqdycxpbyax
dcba
A
Adqbp
X
Aqcpa
Y
Example :
Solve the system by using Cramer’s Rule 8x+5y=2 2x-4y=-10
42410
52
x 4210228
y
142
4242
)50(842
41052
x 24284
42480
4210228
y