Question:
If $A$ and $B$ are two square matrices of the same order such that $B=-A^{-1} B A$, then $(A+B)^{2}=$
Solution:
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, $(A+B)^{2}=\underline{A}^{2}+B^{2}$.
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