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Identity Matrix

For a square ...

Question

For a square matrix $A,$ if $A^{3} = O,$ then $I + A + A^{2}$ is equal to

I + A

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(I + A)^{- 1}

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I - A

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(I - A)^{- 1}

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Solution

The correct option is D $(I - A)^{- 1}$
$A^{3} = O$
$\Rightarrow I - A^{3} = I$
$\Rightarrow (I - A) (I + A + A^{2}) = I$
$\Rightarrow I - A$ and $I + A + A^{2}$ are inverse of each other.
$∴ I + A + A^{2} = (I - A)^{- 1}$

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