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Combination

If A is a nil...

Question

If $A$ is a nilpotent matrix of index $2,$ then for any positive integer $n, A (I + A)^{n}$ is equal to

A

A^{- 1}

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B

A

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C

A^{n}

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D

I_{n}

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Solution

The correct option is B $A$
Given that $A$ is a nilpotent matrix of index $2$
$A^{2} = O, A^{3} = A^{4} = \dots = A^{n} = O$

Then, $A (I + A)^{n} = A (I + n A) = A + n A^{2} = A$

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