If AB=A and BA=B, then which of the following is/are true? (1) A is idempotent (2) B is idempotent (3) At is idempotent (4) None of the above

Answer: (3)

Given

AB=A and BA=B

Consider AB = A

Then A(BA) = A (Since B = BA)

⇒ (AB)A = A

⇒ AA = A (Since AB = A)

⇒ A2 = A

∴ A is an idempotent matrix.

Consider BA = B

Then B(AB) = B (Since A = AB)

⇒ (BA)B = B

⇒ BB = B (Since BA = B)

⇒ B2 = B

∴ B is an idempotent matrix.

Consider A = AB

Applying transpose on both sides

AT = (AB)T

⇒ AT = ATBT………………..(1)

Consider B = BA

Applying transpose on both sides

BT = (BA)T

⇒ BT = BTAT………………..(2)

From equation (1) and (2)

(AT)2 = AT and
(BT)2 = BT

∴ AT and BT are also idempotent matrices.

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