Question:
If $A$ and $B$ are square matrices such that $B=-A^{-1} B A$, then $(A+B)^{2}=$
(a) 0
(b) $A^{2}+B^{2}$
(c) $A^{2}+2 A B+B^{2}$
(d) $A+B$
Solution:
(b) $A^{2}+B^{2}$
$B=-A^{-1} B A$
$\Rightarrow A B=-A A^{-1} B A$
$\Rightarrow A B=-B A$ ....(1)
$\left(\because A A^{-1}=I\right)$
Now,
$(A+B)^{2}=(A+B)(A+B)$
$\Rightarrow(A+B)^{2}=A^{2}+A B+B A+B^{2}$
$\Rightarrow(A+B)^{2}=A^{2}-B A+B A+B^{2}$ [Using (1)]
$\Rightarrow(A+B)^{2}=A^{2}+B^{2}$