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CBSE Class 12 Math 2008 Solved Paper

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Question : 13 of 29
 
Marks: +1, -0
Let A = |325413067|. Express A as sum of two matrices such that one is symmetric and the other is skew symmetric.
OR
If A = |122212221|, verify that A2 - 4A - 5I = 0
Solution:
A = [325413067]
A' = [340216537]
Now, A can be written as:
A = 12 A + A' + 12 A - A'
A + A' = [325413067]+[340216537]
= [3+32+45+04+21+13+60+56+37+7]
= [6656295914]
12 A + A' = 12[6656295914] = [3352319252927] = P, say
Now, P' = [3352319252927]
Thus, P = 12 A + A' is a symmetric matrix.
Now, A - A' = [3−32−45−04−21−13−60−56−37−7] = [0−2520−3−530]
12 A - A' = [0−15210−32−52320] = Q, say
Now, Q' = [01−52−103252−320] = - [0−15210−32−52320] = - Q
Thus, Q = 12 A - A' is a skew symmetric matrix.
∴ A = [325413067] = [3352319252927] + [0−15210−32−52320]
OR
A2 = [122212221] [122212221]
=
[1×1+2×2+2×22×1+2×1+2×21×2+2×2+2×12×1+1×2+2×22×2+1×1+2×22×2+1×2+2×12×1+2×2+1×22×2+2×1+1×22×2+2×2+1×1]

=
[1+4+42+2+42+4+22+2+44+1+44+2+22+4+24+2+24+4+1]
= [988898889]
4A = 4[122212221] =
[1×42×42×42×41×42×42×42×41×4]
= [488848884]
5I = 5 [100010001] = [500050005]
A2 - 4A - 5I = [988898889] - [488848884] - [500050005]
=
[9−4−58−8−08−8−08−8−09−4−58−8−08−8−08−8−09−4−5]

= [000000000] = 0 = R.H.S.
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