Question

If $A$ and $B$ are two square matrices such that $B=-A^{-1}BA,$ then $(A+B{)}^2$ is equal to

Answer 1

As $B=-A^{-1}BA,$
$\Rightarrow AB=-BA\Rightarrow AB+BA=O$

Now,
$(A+B{)}^2=(A+B\left)\right(A+B)$
$=A^2+AB+BA+B^2$
$=A^2+O+B^2=A^2+B^2$