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Q.

If matrix $M = [\begin{matrix} 1 & 0 & 0 \\ 2 & 2 & 1 \\ 0 & 0 & 1 \end{matrix}]$ and let $M_{1} a n d M_{2}$ be two matrices such that $M_{1} = M^{4} - 4 M^{3} + 5 M^{2},$ $M_{2} = M^{4} + 5 M^{2} - 2 M$ then $d e t (M_{1} M_{2})$ is equal to

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a

$2^{11}$

b

$2^{4}$

c

$2^{7}$

d

$2^{13}$

answer is C.

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Detailed Solution

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$I f {(M - I)}^{2} (M - 2 I) = 0 \Rightarrow M^{3} - 4 M^{2} + 5 M - 2 I = 0$

$\Rightarrow M_{1} = 2 M, M_{2} = 4 M^{3} \Rightarrow | M_{1} M_{2} | = | 8 M^{4} | = 11^{4}$

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If matrix M=[100221001] and let M1 and M2 be two matrices such that M1= M4−4M3+5M2, M2= M4+5M2−2M then det(M1M2) is equal to