Question

If A and B are square matrices such that B = − A⁻¹ BA, then (A + B)² =
(a) O
(b) A² + B²
(c) A² + 2AB + B²
(d) A + B

Answer 1

(b) $A^2+B^2$

$$\begin{gathered} B=-A^{-1}BA \\ \Rightarrow AB=-A{A}^{-1}BA \\ \Rightarrow AB=-BA...\left(1\right) \\ \left(\because A{A}^{-1}=I\right) \\ \mathrm{Now}, \\ {\left(A+B\right)}^2=\left(A+B\right)\left(A+B\right) \\ \Rightarrow {\left(A+B\right)}^2=A^2+AB+BA+B^2 \\ \Rightarrow {\left(A+B\right)}^2=A^2-BA+BA+B^2\left[\mathrm{Using}\left(1\right)\right] \\ \Rightarrow {\left(A+B\right)}^2=A^2+B^2 \end{gathered}$$