If A is a square matrix such that $A^2 = A$ and $B=I-A$, then $AB + BA + I-(I-A)^2 =$
Answer & explanation
Correct answer: option 1
We have,
$A^2= A$ and $B = I-A$
$∴AB + BA+I-(I-A)^2$
$=A (I-A) + (I-A) A+ I-(I-A) (I - A)$
$=A-A^2+A-A^2 + I -(I-2A + A^2)$
$=A- A+ A- A+ I-(I-2A + A)$ $[∵ A^2=A]$
$= A$