If $A$ is a square matrix such that $A^{2} = A$ and $B = I - A$ , then $A B + B A + I - (I - A)^{2}$ is equal to.

A

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2 A

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- A

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I - A

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Solution

The correct option is C $A$
We have, $A^{2} = A$ and $B = I - A$
$∴ A B + B A + I - (I - A)^{2}$
$= A (I - A) + (I - A) A + I - (I - A) (I - A)$
$= A - A^{2} + A - A^{2} + I - (I - 2 A + A^{2})$
$= A - A + A - A + I - (I - 2 A + A)$ $[∵ A^{2} = A]$
$= A$ .

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