If A is a square matrix such that $A^2 = A$ and $B=I-A$, then $AB + BA + I-(I-A)^2 =$

$A$ $2A$ $-A$ $I-A$

$A$

We have, $A^2= A$ and $B = I-A$ $∴AB + BA+I-(I-A)^2$ $=A (I-A) + (I-A) A+ I-(I-A) (I - A)$ $=A-A^2+A-A^2 + I -(I-2A + A^2)$ $=A- A+ A- A+ I-(I-2A + A)$   $[∵ A^2=A]$ $= A$