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Operations with Matrices (i, j)-th entry: row: m column: n size: m×n
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Operations with Matrices Properties of Matrix Operations The Inverse of a Matrix
Mrs. Meena KumariDEPARTMENT OF ECONOMICSPGGCG-11 ,CHANDIGARH
MATRICES
nm
nmmnmmm
n
n
n
ij M
aaaa
aaaaaaaaaaaa
aA
][
321
3333231
2232221
1131211
Operations with Matrices
(i, j)-th entry:
ija
row: mcolumn: n
size: m×n
i-th row vector
iniii aaar 21 row matrix
j-th column vector
mj
j
j
j
c
cc
c2
1
Square matrix: m = n
Diagonal matrix:
),,,( 21 nddddiagA nn
n
M
d
dd
00
0000
2
1
Trace:
nnijaA ][ If
nnaaaATr 2211)(Then
Example:
654321
A
2
1
rr
,3211 r 6542 r
654321
A 321 ccc
,41
1
c ,
52
2
c
63
3c
Equal matrix:
nmijnmij bBaA ][ ,][ If
njmibaBA ijij 1 ,1 ifonly and if Then
Example: (Equal matrix)
dcba
BA 4321
BA If
4 ,3 ,2 ,1 Then dcba
Matrix addition:
nmijnmij bBaA ][ ,][ If
nmijijnmijnmij babaBA ][][][Then
Example : (Matrix addition)
3150
21103211
2131
1021
231
231
2233
11
000
Scalar multiplication:
scalar : ,][ If caA nmij
nmijcacA ][Then
Matrix subtraction:
BABA )1(
Ex 3: (Scalar multiplication and matrix subtraction)
212103421
A
231341002
B
Find (a) 3A, (b) –B, (c) 3A – B
Sol:(a)
212103421
33A
231323130333432313
636309
1263
(b)
231341002
1B
231341002
(c)
231341002
636309
12633 BA
4076410
1261
Matrix multiplication:
pnijnmij bBaA ][ ,][ If
pmijpnijnmij cbaAB ][][][Then
Size of ABwhere
njin
n
kjijikjikij babababac
1
2211
inijii
nnnjn
nj
nj
nnnn
inii
n
ccccbbb
bbbbbb
aaa
aaa
aaa
21
1
2221
1111
21
21
11211
BAAB
Example : Show that AB and BA are not equal for the matrices.
1231
A and
2012
B
Sol:
4452
2012
1231
AB
2470
1231
2012
BA
Note: Note: BAAB
Ex : (Find AB)
052431
A
1423
B
Sol:
)1)(0()2)(5()4)(0()3)(5()1)(2()2)(4()4)(2()3)(4()1)(3()2)(1()4)(3()3)(1(
AB
10156419
Three basic matrix operators: (1) matrix addition (2) scalar multiplication (3) matrix multiplication
Properties of Matrix Operations
Zero matrix:nm0
Identity matrix of order n:nI
(1) A+B = B + A
Properties of matrix addition and scalar multiplication:
(2)A + ( B + C ) = ( A + B ) + C
(3) 1A = A
(4) c( A+B ) = cA + cB
calar cMA nm s: , If
Properties of zero matrices:
AA nm 0 (1)Then
nmA A 0)((2)
nmnm or A c cA 000)3(
Notes:
(1) 0m×n: the additive identity for the set of all m×n matrices(2) –A: the additive inverse of A
Transpose of a matrix:
nm
mnmm
n
n
M
aaa
aaaaaa
A
If
21
22221
11211
mn
mnnn
m
m
T M
aaa
aaaaaa
A
Then
21
22212
12111
Transpose of A matrix
(a)
82
A (b)
987654321
A (c)
114210
A
Sol: (a)
82
A 82 TA
(b)
987654321
A
963852741
TA
(c)(c)
114210
A
141
120TA
Properties of transposes:
AA TT )( )1(TTT BABA )( )2(
)()( )3( TT AccA
)( )4( TTT ABAB
Symmetric matrix:
A square matrix A is symmetric if A = AT
Skew-symmetric matrix:
A square matrix A is skew-symmetric if AT = –A
Example:
654321
Ifcb
aA is symmetric, find a, b, c?
Sol:
654321
cbaA
65342
1cba
AT
5 ,3 ,2 cba
TAA
Ex:
030210
Ifcb
aA is a skew-symmetric, find a, b, c?
Sol:
030210
cbaA
032
010
cba
AT
TAA 3 ,2 ,1 cba
The Inverse of a Matrix
The inverse of a matrix is unique.
(1) The inverse of A is denoted by
1A
IAAAA 11 )3(
AAA 111 )( and invertible is (2)
If A and B are invertible matrices of size n, then AB is invertible and
111)( ABAB