Question

If A and B are two matrices such that AB = A and BA = B, then B² is equal to
(a) B
(b) A
(c) 1
(d) 0

Answer 1

(a) B

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