Question

If A and B are two matrices such that AB = B and BA = A, A² + B² is equal to
(a) 2 AB
(b) 2 BA
(c) A + B
(d) AB

Answer 1

(c) A + B

$$\begin{gathered} \mathrm{Given}:AB=B\mathrm{and}BA=A \\ {A}^2+B^2=AA+BB \\ \Rightarrow A^2+B^2=BABA+ABAB\left[\because AB=B\mathrm{and}BA=A\right] \\ \Rightarrow A^2+B^2=BBA+AAB\left[\because AB=B\mathrm{and}BA=A\right] \\ \Rightarrow A^2+B^2=BA+AB\left[\because AB=B\mathrm{and}BA=A\right] \\ \Rightarrow A^2+B^2=A+B\left[\because AB=B\mathrm{and}BA=A\right] \end{gathered}$$