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Question

If $$A$$ and $$B$$ are two matrices such that $$AB = B, BA = A$$ then $$[A$$ and $$B$$ are not null matrices].


A
A2=A
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B
B2=0
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C
AB is idempotent
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D
AB is nilpotent
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Solution

The correct option is A $$A^{2} = A$$
$$AB=B$$  $$\&$$  $$BA=A$$
$$\Rightarrow AB=B$$
$$\Rightarrow (BA)B=B$$
$$\Rightarrow B(AB)=B$$
$$\Rightarrow B.B=B$$
$$\Rightarrow B^2=B$$
Similarly, $$A^2=A$$
Hence, the answer is $$A^2=A.$$

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