If A and B are two square matrices of the same order

Given:

$B=-A^{-1} B A$

$\Rightarrow A B=-A A^{-1} B A$

$\Rightarrow A B=-I B A$

$\Rightarrow A B=-B A$

Now,

$(A+B)^{2}=A^{2}+A B+B A+B^{2}$

$=A^{2}-B A+B A+B^{2} \quad(\because A B=-B A)$

$=A^{2}+B^{2}$

Hence, $(A+B)^{2}=\underline{A}^{2}+B^{2}$.