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OBJECTIVES
At the end of the lesson, students shouldbe able to:
(c) perform operations on matrices such
as multiplication of two matrices
(d) define the transpose of a matrix and
explain its properties
(b) define symmetric matrix and skew
symmetric matrix
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Multiplication of Matrices
The product of two matrices A and B is
defined only when the number of
columns in Ais equal to the number ofrows in B.
If order of A is mx nand the order of Bis nxp, then ABhas order mx p.
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The order of the
product is mx p
These numbers must
be equal
m
n np
AmxnBnxp ABmxp
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A row and a column musthave thesame number of
entriesin order to be
multiplied.
ATTENTION !
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Multiplication Of Two Matrices
naaaaR
321
nb
b
bb
C
3
2
1
1 1 2 2 3 3[ ]
n nRC a b a b a b a b
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Find :
12
43
12
502
321
Example 1
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12
43
12
502
321
)1(5)4(0)1(2)2(5)3(0)2(2
)1(3)4(2)1(1)2(3)3(2)2(1
36
122
Solution
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Find AB.
2 5 4
1 7 5
A
1 2 3 5
3 2 1 5
5 4 0 7
B
Example 2
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7 30 11 4345 4 10 5
AB
Solution
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Let and
Show that AB BA .
43
21A
23
12B
Example 3
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43
21
23
12
1 2 2 3 1 1 2 2
3 2 4 3 3 1 4 2
4 5
18 5
AB =
Solution
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BA =
=
=
2312
4321
)4(2)2(3)3(2)1(34)1()2(23)1()1(2
143
05
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Thus, AB BA .
This result prove that the matrixmultiplication is not commutative.
51854
14305
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Properties of Matrix Multiplication
Let A,B,C and D be matrices for which
the following products are defined .Then
ASSOCIATIVE PROPERTY
A(BC) = (AB)C DISTRIBUTIVE PROPERTY
A(B+C) = AB+AC
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Transpose Matrix
Definition
The transpose of a matrix A , written as AT
,is the matrix obtained by interchanging the
rows and columnsof A . That is, the
ith
column of AT
is theithrow ofA for all is.
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If Amxn= [aij] ,
then ATnxm= [aji]
11 12 13
21 22 23
31 32 33 3 3
a a a
A a a a
a a a
33332313
322212
312111
aaa
aaa
aaa
AT
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531
452
331
DIf then T =
543
353
121
133
1
2
BT
BLet then 31312
Example 4
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Let , and
Show that
(a) (A + B)T = A T+ BT
(b) (BC)T = CTBT
1 2
3 4A
3 4
2 1B
1 4
3 2C
Example 5
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(a) A + B =
1423
4231
55
64
(A + B)T =
56
54
=
Solution
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14
23
42
31
=
56
54
(A + B)T = AT+ BT
AT+ BT =
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) BC =
3 4 1 4
2 1 3 2
=
2822 =
25
[BC]T =
55
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Properties of transpose
(A B)T = AT BT(AT)T = A
(AB
)
T=
BTA
T
(kA)T = kAT ,kis a scalarTT
BA
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A square matrix,
A = [ aij]
nxn,
is symmetric if it is equal to its own
transpose,A = A T that is aij= aji
A Symmetric Matrix
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32
21,
32
21T
AA
A and B are symmetric matrices.
2
3
1
,
2
3
1
cbca
ba
Bcb
ca
ba
B
T
Example 6
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Given . Find AT.
Hence, prove that AATis a symmetric
matrix.
43
21A
Example 7
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42
31
43
21 T
TA
TAA
42
31
43
21
2511
115
Solution
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(AAT)T =
T
2511115
2511
115
= AAT
Since AAT= (AAT)T,
therefore AAT is a
symmetrical matrix
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A Skew Symmetric Matrix
A square matrix,
A = [ aij]nxn,
is a skew symmetric matrix if A = -A T
jiij aa where i j and 0iia
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02
20,
02
20TAA
A and B are skew symmetric matrix.
031
302
120
,
031
302
120
TBB
02
20A
B
Example 8
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A Symmetrical Matrix A = AT that is aij = aji
A = -AT, jiij aa A Skewed Symmetrical Matrix
Transpose Matrix
Properties of transpose
(A B)T = AT BT(AT)T = A(AB)T = BTAT(kA)T = kAT ,kis a scalar
TTBA
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Multiplication Of Two Matrices
naaaaR
321
nb
b
bb
C
3
2
1
1 1 2 2 3 3[ ]
n nRC a b a b a b a b
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032
421A
43
21B
462154
973
C
1. Let
,
and
Indicate whether the given product is defined.
If so, give the order of the matrix product.Compute the product, if possible.
(a) AB (b) AC (c) BA
(d) BC (e) CA (f) CB
Exercises:
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12
04
13
A
11
21
12
B
22
43C
2. Let
,
and
Find , (a) ATB (b) BTA
(c) (BC)T
(d) (A+B)T
A
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1.(a) Not defined
(b) Defined ; 2 x 3
(c) Defined ; 2 x 3
(d) Not defined.
(e) Not defined.
f Not defined
212918
274119
121811
485
Answers:
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01
1312
013
112
6810578
220
355
2 (a) (b)
(c) (d)