Question:
If $A B=A$ and $B A=B$, where $A$ and $B$ are square matrices, then
(a) $B^{2}=B$ and $A^{2}=A$
(b) $B^{2} \neq B$ and $A^{2}=A$
(c) $A^{2} \neq A, B^{2}=B$
(d) $A^{2} \neq A, B^{2} \neq B$
Solution:
(a) $B^{2}=B$ and $A^{2}=A$
Here,
$A B=A \quad \ldots(1)$
$B A=B \quad \ldots(2)$
$\Rightarrow A B A=A A \quad$ [Multiplying both sides by $A$ ]
$B A B=B B$ [Multiplying both sides by $A$ ]
$\Rightarrow A B=A^{2} \quad[$ From eq. (2) $]$
$B A=B^{2} \quad$ [From eq. (1) $]$
$\Rightarrow A=A^{2} \quad[$ From eq. $(1)]$
$B=B^{2} \quad$ [From eq. (2)]
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