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Question

If AB=A and BA=B, then which of the following is/are true?

A
A is idempotent
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B
B is idempotent
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C
AT is idempotent
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D
none of these
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Solution

The correct options are A A is idempotent B B is idempotent C AT is idempotentGiven, AB=A and BA=BTake, AB=A⇒A(BA)=A [ Since, B=BA ]⇒(AB)A=A ⇒AA=A [ Since, AB=A ]⇒A2=ATherefore, A is an idempotent matrix.Take, BA=B⇒B(AB)=B [ Since, A=AB ]⇒(BA)B=B⇒BB=B [ Since, BA=B ]⇒B2=BTherefore, B is an idempotent matrix.Now, A=ABApplying transpose on both sides, we getAT=(AB)T⇒AT=BT.AT....(i)Also, B=BAApplying transpose on both sides, we getBT=(BA)T⇒BT=AT.BT...(ii)From equation (i) and equation (ii), we get(AT)2=AT and(BT)2+BTTherefore, AT and BT are alos idempotent matrices.Hence, the correct options are (A),(B) and (C).

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