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Mat.ematics
Matrices
/+No+1Find the order of matrices C= 2 4 Order = m x n
A=[ 2 35 6] No. of rows = 2 No. of rows = M = No. ofrows
No. of col!mns = 2 No. of Col!mns = 2 n = No. of col!mnsOrder = 2 x 2 O" 2 by 2 Order = x 2
# =
[4
0
6 ]$ =
[a d
b ec f]
F = [ 2 ]
% = [2 3 01 2 32 4 5
]Order = & x Order = & x 2 Order = x
Order = & x &
' = [2 3 41 0 6 ] Order of [ 0 ] ([0 00 0 ],[00] )))))))))))*Order = 2 x &
/+No+ 0Matrices are e+!al if order and corresponding entry is same,
- = [3 5 ] 2 ( / = [3 3+ 2 ] 2are e+!al i.e -=/
C = = [5 2 ] A = [ 3 ] C =A
$ = [4 06 2] 2 x2 ( 0 = [2+2 222+ 4 2+0 ] 2 x2 ( $ = 0 = '( ' = [4 06 2] 2 x2# is not e+!al to any gi1en matrix.
F = [26] 2 x2 ( % = [313+3 ] 2 x F=%
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/+No+ (al!e of a ( b *
[a+c a+2b
c1 4 d6 ]2 x2 =
[0 7
3 2 d
]2 x2
-y comparing corresponding entries.
a 3 c = 4 i a 3 2b = )5 ii
c) = 4 6d )7 = 2d
6d )2d = 37
i gi1es 2d = 7
a 3 89 = 4 d =6
2
$+ ii becomes
) 32b = )5
2b = )53
2b = )7
2b =6
2
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
E+ 1+0
Name the matrices.
A = [0 00 0 ] N!ll matrix d = [1 00 1 ] ;nit matrix
- =
2 3 4 &
row matrix $ = [1 0 0 ] [ 1 1 1 ]
C=
d= &a=
b=
N!ll Matrix"ow MatrixCol!mn Matrix
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[0 0 0 ]
[
0
0
0
]
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/+No+ 0Name the matrices
i.
[8 2 712 0 4
] "ectangle matrix
ii. [301] & Col!mn Matrix
"ectangle Matrix
1. [1 23 45 6
] & "ectang!lar Matrix1ii. [10
0] & "ectangle matrix
Col!mn Matrix
ix. [0 00 00 0
] & N!ll Matrix "ectang!lar Matrix
iii.
[6 43 2 ]
2 x2 ?+!are Matrix
i1. [1 00 1 ] 2 x2 ?+!are Matrix /dentity Matrix
1i. 3 101 x & "ow Matrix
"ectang!lar
Matrix
1iii. [1 2 31 2 00 0 1
] x & ?+!areMatrix
[5 ] *** [ 1 ]
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/+No+(
A.
[4 0
0 4 ] #iagonal Matrices
?calar Matrices
-. [2 00 1] #iagonal Matrices
C.[1 00 1 ] #iagonal Matrix ;nit or /dentity
#. [3 00 0 ] #iagonal Matrix#iagonal Matrix
$. [53 00 1+1] ?calar Matrix/+No+$
Negati1e of gi1e matrices
A 2 [ 101
] )A = [ 101
] = 8say9)- = [ 3 +12 1] ( )C = [2 63 2]
)# =
[+3 2
+4 1]( )$ =
[1 52 3
]O = [00] * 8Negati1e9
= [0 00 0 ]
/+No+3
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!o*s in o"ms o column *ill e t"anspo"t o mat"i+
At2
[ 0
1
2]t= [ 0 12 ] x&(
Pt2 [ 3 21 1] 2x2= [3 12 1] t2x2(
Bt 2 [5 16 ] tx&=
[
5
1
6
]&x( Ct =
[
1 2
2 1
3 0
]&x22 [1 232 10 ] 2x&
#t=[2 03 5] ( $t= * ( Ft= */+No+4
A 2[1 20 1] ( - 2 [1 12 0]
At=
[1 0
2 1]( -t=
[1 2
1 0]8At9t= [1 20 1] ( 8-t9t= [1 12 0]
1+(
/+No+1
/f order of matrices is same then they can be added to form a new matrix.A( $ Can be added 8Order 2x29
-( # Can be added 8Order 2x29
C( F Can be added 8Order &x29
/+No+0
/f s!m of two matrices is small matrix of same order then they are additi1e in1erse
of each other.
%t2 [2 36 2] 2x2(
t= [3 4
2 5]2x2(
8-t9t=8At9t=
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Additi1e in1erse of A2 [ 2 42 1 ] is [2 4+2 1] = Aor BA 8say9A3A = 4 A( Aare additi1e in1erse of each other.
2 [ 2 42 1 ]+[2 42 1 ] = [ 22 442+2 11 ] = [0 00 0 ] 2x21eried
/n1erse of - = [1 0 12 1 33 2 1] is [
1 0 +12 +1 33 +2 1]
/n1erse of C =
[ 4
2]is
[4+2 ]
/n1erse of # = [ 1 03 22 1
] is [1 0+3 +22 1]/n1erse of $ = [1 00 1 ] is [1 00 1]
/n1erse of F =
[3 1
1 2]is
[3 1
+1 2]For addition and s!bstit!tion order of both matrices sho!ld be same.
i+
A3 [1 11 1]
= [1 22 1] 3 [1 11 1]
= [1+1 2+12+1 1+1]
= [0 33 2]ii+
-3[23]
i5+
# 3 [0 1 02 0 1]
= [ 1 2 31 0 2] 3 [0 1 02 0 1]
= [ 1+0 2+1 3+01+2 0+0 2+1]
= [1 3 31 0 3]5+
2A
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= [ 11] 3 [23]
= [ 1 21 +3 ]
= [12]iii+
C3 [2 1 3 ]
= [1 1 2 ] 3 [2 1 3 ]
= 12 1+1 2+3
= [1 0 5 ]
= 2 [1 22 1]
= [1X2 2X22X2 1X2 ]
=[1 44 2 ]5i+
8)9-
= 8)9
[ 1
1]=
1X11 X(1)
= [1+1 ]1ii.
)2C
=))2 1 1 2
=
= [2X1 2X(1) 2X2 ]
= [2 +2 4 ]
1iii.
( ix9 &C( A3- ))) *
/+No+$
?ol1e yo!rself C3#))))))))*
([1 00 1]+[0 23 0])+[1 11 0 ]
= [1 23 1] 3 [1 11 0]
i1.
[ 1 2 3
1 1 1
0 1 2]3
[1 1 1
2 2 2
3 3 3]= [ 1+1 2+1 3+11+2 1+2 1+2
0+3 1+3 2+3]
= [2 3 41 1 13 4 5
]1.#.D.?( 1i. #.D.?
/+No+3
A3C = C3A
E.'.? = A3C
= [1 2 32 3 11 1 0
] 3 [1 0 00 2 31 1 2
]
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= [2 34 1]ii.
#.D.?
iii.
1
[2 3 1 ]+ ([ 02] [2 2 2])
= [2 3 1 ] [ 12 02 22 ]
= 2 3 1 3 12 02 22
= [2 3 1 ] 3 [0 1 0 ]
= 2 2 1
#.D.?
A38-3A9=2A3-
E.'.? = A38-3A9
= [1 2 3
2 3 1
3 1 0] 3 ([1 1 12 2 23 1 3]+[
1 2 3
2 +2 11 1 0])
= [1 2 32 3 11 1 0
] 3 [2 +1 44 +1 34 0 3
]".'.? = 2A 3-
= 2 [1 2 3
2 3 1
1 1 0] 3 [1 1 12 2 23 1 3]
= [2 4 64 6 22 2 0
] 3 [1 1 12 2 23 1 3
]".'.? A-O$
E.'.? = ".'.? "O$# #.D.?
5ii+
E.'.? =
[
0 2 3
2 1 4
2 0 2
]".'.? = C3A= [1 0 00 2 3
1 1 2] 3 [1 2 32 3 1
1 1 0]
".'.? = [0 2 32 1 42 0 2
]From abo1e
E.'.? = ".'.? pro1ed
> C!m!lati1e property w.r.t G3H
. 8C)-93A = C38A)-9
E.'.? = 8C)-93A
=
([
1 0 0
0 2 3
1 1 2
]+
[
1 1 1
2 2 2
3 1 3
])3A
= [ 0 1 12 0 12 0 1
] 3 [1 2 32 3 11 1 0
]E.'.? = [ 1 3 20 3 2
1 1 1]
".'.? = [1 3 2
0 3 2
1 1 1]F"OM A-O$
E.'.? = ".'.? "O$#
> 2)8)9 = 23=&
)2)8)29 = )232=4
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8C)-9)A = 8C)A9)-
E.'.? = 8C)-9)A
= ([1 0 0
0 2 31 1 2]+[
1 1 12 2 23 + 1 3])A
= [2 +1 12 + 03 121 0 1] ) [1 2 3
2 3 1
1 1 0]E.'.? =
[3 1 4
4 3 03 +1 1]".'.? = 8C)A9)-
= ([1 0 00 2 31 1 2
]+[1 2 32 3 11 1 0])B
=
[2 2 32 5 2
0 2 2
])
[1 1 12 2 23 1 3
]".'.? = [3 1 44 3 0
3 1 1]
F"OM A-O$
E.'.?=".'.? "O$#
5iii+
i e 5e"i6ed easily
2A = 2
[
1 2 3
2 3 1
1 1 0
]=
[
2 4 6
4 6 2
2 2 0
]2-= [1 1 12 2 2
3 1 3] = [2 2 24 4 4
6 2 6]
E.'.? = 2A32-
= [2 4 64 6 22 2 0
] 3 [2 2 24 4 46 2 6
]E.'.?= [4 2 88 2 6
8 0 6]
A3- = [1 2 32 3 11 1 0] 3 [
1 1 12 2 23 1 3
]A3- =
[2 1 4
4 1 3
4 0 3 ]
28A3-9= 2 [2 1 44 1 34 0 3
]".'.? =
[4 2 8
8 2 68 0 6]
E.'.? = ".'.? "O$#
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