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Inverse of a Matrix

If A3=O, then...

Question

If $A^{3} = O$ , then $I + A + A^{2}$ equals to

I - A

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(I + A)^{- 1}

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(I - A)^{- 1}

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I + A

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Solution

The correct option is C $(I - A)^{- 1}$
Given $A^{3} = O$
Now, consider $(I - A) (I + A + A^{2}) = I^{2} + I A + I A^{2} - A I - A^{2} - A^{3}$
$\Rightarrow (I - A) (I + A + A^{2}) = I$
$\Rightarrow (I - A)^{- 1} = I + A + A^{2}$

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