9th Math(Matrices)

7/25/2019 9th Math(Matrices)

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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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9th Math(Matrices)