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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 idempotent
Given, AB=A and BA=B

Take, AB=A
A(BA)=A [ Since, B=BA ]
(AB)A=A
AA=A [ Since, AB=A ]
A2=A
Therefore, A is an idempotent matrix.

Take, BA=B
B(AB)=B [ Since, A=AB ]
(BA)B=B
BB=B [ Since, BA=B ]
B2=B
Therefore, B is an idempotent matrix.

Now, A=AB
Applying transpose on both sides, we get
AT=(AB)T
AT=BT.AT....(i)

Also, B=BA
Applying transpose on both sides, we get
BT=(BA)T
BT=AT.BT...(ii)

From equation (i) and equation (ii), we get
(AT)2=AT and
(BT)2+BT

Therefore, AT and BT are alos idempotent matrices.

Hence, the correct options are (A),(B) and (C).

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