Question

If AB = A and BA = B, where A and B are square matrices, then
(a) B² = B and A² = A
(b) B² ≠ B and A² = A
(c) A² ≠ A, B² = B
(d) A² ≠ A, B² ≠ B

Answer 1

(a) B² = B and A² = A

$$\begin{gathered} \mathrm{Here}, \\ AB=A...\left(1\right) \\ BA=B...\left(2\right) \\ \Rightarrow ABA=AA\left[\mathrm{Multiplying}\mathrm{both}\mathrm{sides}\mathrm{by}A\right] \\ BAB=BB\left[\mathrm{Multiplying}\mathrm{both}\mathrm{sides}\mathrm{by}A\right] \\ \Rightarrow AB=A^2\left[\mathrm{From}\mathrm{eq}.\left(2\right)\right] \\ BA=B^2\left[\mathrm{From}\mathrm{eq}.\left(1\right)\right] \\ \Rightarrow A=A^2\left[\mathrm{From}\mathrm{eq}.\left(1\right)\right] \\ B=B^2\left[\mathrm{From}\mathrm{eq}.\left(2\right)\right] \end{gathered}$$