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If A & B are two matrices such that AB = B, BA = A then

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${(A + B)}^{10} = 2^{11} (A + B)$

${(A + B)}^{10} = {(A + B)}^{11}$

${(A + B)}^{10} = 2^{10} (A + B)$

${(A + B)}^{11} = 2^{10} (A + B)$

answer is C.

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Detailed Solution

AB = B, BA = A $\Rightarrow$ A & B idempotent

$A^{2} = A, B^{2} = B$

${(A + B)}^{2} = A^{2} + A B + B A + B^{2} = 2 (A + B)$

${(A + B)}^{3} = {(A + B)}^{2} (A + B) = 2 (A + B) (A + B) = 2 {(A + B)}^{2} = 2.2 (A + B)$ so on.

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