Question

If A and B are two square matrices of the same order such that B = -A^-1BA, then (A+B)² = ______________.

Answer 1

Given:
$$\begin{gathered} B=-A^{-1}BA \\ \Rightarrow AB=-A{A}^{-1}BA \\ \Rightarrow AB=-IBA \\ \Rightarrow AB=-BA \end{gathered}$$


$$\begin{gathered} \mathrm{Now}, \\ {\left(A+B\right)}^2=A^2+AB+BA+B^2 \\ =A^2-BA+BA+B^2\left(\because AB=-BA\right) \\ =A^2+B^2 \end{gathered}$$


Hence, (A + B)² = A² + B².