Question

If $A$ and $B$ are two matrices such that $AB=B$ and $BA=A$, then $A^2+B^2$ equals

• A. $2AB$
• B. $2BA$
• C. $A+B$
• D. $AB$

Answer 1

The correct option is C $A+B$

It is given that $AB=B$ and $BA=A$
Then, $A^2+B^2=A.A+B.B$
$=A\left(BA\right)+B\left(AB\right)$
$=\left(AB\right)A+\left(BA\right)B$
$=BA+AB$
$=A+B$