Question:
If $A$ and $B$ are two matrices such that $A B=A$ and $B A=B$, then $B^{2}$ is equal to
(a) $B$
(b) $A$
(c) 1
(d) 0
Solution:
(a) $B$
Here,
$A B=A \quad \ldots(1)$
$B A=B \quad \ldots(2)$
$\Rightarrow B A B=B B \quad[$ Multiplying both sides by $B]$
$\Rightarrow B A=B^{2} \quad$ [From eq. (1)]
$\Rightarrow B=B^{2} \quad$ [From eq. (2)]
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