Question

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

Answer 1

$B=-A^{-1}BA⟹AB=-A{A}^{-1}BA=-BA$
Since $A{A}^{-1}=I$ ....... (1)
$\therefore$ AB + BA = 0
Now $(A+B{)}^2=(A+B\left)\right(A+B)$
= AA + (AB + BA) + BB = $A^2+B^2$ by (1)