8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
1/42
5.0 MATRICES AND SYSTEM OF
LINEAR EQUATIONS
5.1 MATRICES
5.2 DETERMINANT OF MATRICES
5.3 INVERSE MATRICES
5.4 SYSTEM OF LINEAR EQUATIONSWITH THREE VARIABLES
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
2/42
LECTURE 1 OF 11
5.0 Matrices And Systems OfLinear Equations
5.1 Matrices
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
3/42
At the end of the lesson, students
should be able to:
(a) define matrix and equality of matrices.
(b) identify different types of matrices
such as row, column, zero,
diagonal, upper triangular, lower
triangular and identity matrices.
(c) perform operations on matrices such as
- addition
- subtraction
- scalar multiplication
LEARNING OUTCOMES
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
4/42
The result of EURO 2006
P W D L G PTS
France 3 2 1 0 7 7England 3 2 0 1 8 6
Croatia 3 0 2 1 4 2
Switzerland 3 0 1 2 1 1
Group B
The above standing shows MATRIXform.
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
5/42
Matrix Definition
A matrix is a rectangular array ofnumbersenclosed between brackets.
The general form of a matrix withm rowsand n columns:
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
6/42
mnmmm
n
n
n
aaaa
aaaa
aaaa
aaaa
321
3333231
2232221
1131211
m rows
n columns
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
7/42
The order or dimension of a matrix
withm rowsand n columnsis mxn.
ija
The numbers that makes up a matrix
are called itsentries orelements,
and they are specified by their row
andcolumnposition.
Thematrixfor which the entry is in
ith rowandjthcolumnis denoted by
ija
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
8/42
Let A=
7322
165
(a) What is the order of A?
(b) If A = [ aij] identify a21and a13
Example 1
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
9/42
(a)Since A has 2 rows and 3 columns,
the order of A is 2 x 3.
(b) The entry a21is in the second
row and the first column.
Thus, a21=
The entry a13is in the first row and the
third column, and so a13=2
1
-2
Solution
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
10/42
3 3ija
Given A =
Find matrix A if2
ij
ij i ja
j i i j
Example 2
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
11/42
11a
12a
13a
31a
21a
22a
23a
32
a
33a
1(1) =
1(2) =
2(2) =
1(3) =
2(3) =
3(3) =
1
2(1) + 2 =
2(2) + 3 =
2(1) + 3 =
3
2
4
4
6
9
7
5
975
644321
A
Solution
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
12/42
Two matrices are equal if they have the same
dimensionand their corresponding entries areequal
12
21Which matrices below are the same?
A = , B = , C =
12
12
21 21
D =
1221
Solution: A = D
Equality of Matrices
Example 3
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
13/42
Let A =
248
463
b
a
B=
2832
469
d
c
If A = B, find the value of a, b, cand d.
Example 4
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
14/42
b = -2
c = 0
3 a = 9
a = -6
4b = -8
6 c = 6 2 3d= 8
3d = -6
d = -2
a=-6 , b = -2, c = 0, d= -2
Solution
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
15/42
Types of Matrices
1. Row Matrixis a (1 x n) matrix [one row]
A = [ a11 a12 a13 a1n]
Example
A = [ 1 2 3 ]
B = [ 1 0 7 8 4 3 5 ]
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
16/42
11
21
31
1m
aa
a
.
.
.
a
2. Column Matrixis a (m x 1)
matrix [ one column ]
A =
Example
A =
0
4
,B =
7
5
3
2
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
17/42
3. Square Matrixis a (nxn) matrix which
has the same number of rows ascolumns.
Example
81
31A = , 2 x 2 matrix
B =
132
213
231
, 3 x 3 matrix
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
18/42
4. Zero Matrixis a (m x n) matrix which
every entry is zero, anddenoted by .
Example
000000
000
O = O =
00
00O =
0000
00
O
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
19/42
mmmmm
m
m
m
aaaa
aaaa
aaaa
aaaa
321
3333231
2232221
1131211
5. Diagonal Matrix
Let A =
The diagonal entries of A are a11,a22 ,.,amm
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
20/42
A =
30
02,B =
300
020
001
,C =
b
a
00
000
00
Example
A square matrix which non-diagonal entriesare all zero is called a diagonal matrix
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
21/42
6. Identity Matrix is a diagonal matrix
where all its diagonal entries are 1 anddenoted byI.
10
01
100
010
001
= I2x2= I3x3
Example
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
22/42
7. Lower Triangular Matrixis a square matrix
and aij= 0 for i < j
323
023
001
A = ,B =
edc
fb
a
0
00
333231
232221
131211
aaa
aaa
aaa
Example
8 Upper Triangular Matrix is a square matrix
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
23/42
8.Upper Triangular Matrixis a square matrix
and aij= 0 for i > j
300
420
321
P = R =
f
ed
cba
00
0
333231
232221
131211
aaa
aaa
aaa
Example
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
24/42
Addition And Subtraction Of Matrices
For mx nmatrices
A = and B =
A + B = C = where
A B = D = where
]a[ij
ijijij bac
.ijijij bad
]b[ij
]c[ij
]d[ij
Operations on Matrices
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
25/42
NOTE
The additionor subtraction
of two matrices with differentordersis not defined.
We say the two matrices areincompatible.
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
26/42
43
21A
,
65
34B
2
1C
.
FIND :
(a) A + B (b) A
B
(c) A + C
Example 5
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
27/42
(a) A + B =
43
21 +
65
34
64)5(3
3241
=5 5
2 10
Solution
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
28/42
(b) A
B =
43
21
-
65
34
=
64)5(33241
=
28
13
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
29/42
Since matrix A is of order
2 x 2 and matrix C is of order
2 x 1, the matrices have differentorders, thus A and C are
incompatible.
4321
+ 21c) A + C =
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
30/42
ijcaijb
then
][ ijaA
where]b[cA ijIf cis a scalarand
Scalar Multiplication
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
31/42
2 4
8 5
6 7
A
Given
1
2AFind
Example 6
Solution
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
32/42
1 12 2
1 12 2
1 1
2 2
2 4
1 8 52
6 7
A
2
7
25
3
4
21
=
Solution
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
33/42
Let and
calculate 3A 2B
3541A
2463B
Example 6
Solution
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
34/42
35 413 24 632
48
126
915
123
57
03
=
=
=
Solution
3 2A B
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
35/42
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
36/42
Types of Matrices
1. Row Matrix
2. Column Matrix
3. Square Matrix
4.Zero Matrix
5. Diagonal Matrix
6. Identity Matrix
7. Lower/Upper Triangular Matrix
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
37/42
NOTE
The additionor subtraction
of two matrices with differentordersis not defined.
We say the two matrices are
incompatible.
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
38/42
1. Identify the order of the given matrix
654
321(a) ( b )
10
01
01
(c)
d
c
b
a
(d) kji
Exercises:
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
39/42
2
2
ij
( i j ) ,i j b = ij ,i j
( i j ) ,i j
2(a) Find matrix A = [aij]2x3
if aij= i2j + j2i
(b) Find matrix B = [bij]3x3
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
40/42
53
21A
94
12B
456
321
000
000
42
004
33
62XX
3. Simplify the given quantity for
and
(a) A + B (b) AB
(c) 2A5B (d) 3A + 2B
=
(b) -2
4. Solve the given equation for
the unknown matrix X.
(a) 2X+
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
41/42
2 6 12
6 16 30
1 4 5
0 4 7
1 1 9
1. (a) 2 X 3 (b) 3 X 2(c) 4 X 1 (d) 1 X 3
(b)
2. (a)
ANSWERS
8/12/2019 Lecture 1 of 11 (Chap 5, Matrices)
42/42
147
33
41
11 3. (a) (b)
3514
18
3317
87(c) (d)
23
1
25
23
21
62
4. (a)
(b)