Question

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

• A. A is idempotent
• B. B is idempotent
• C. A${}^T$ is idempotent
• D. none of these

Answer 1

Answer: (A), (B), (C)

The correct options are
A A is idempotent
B B is idempotent
C A${}^T$ is idempotent

Given, $AB=A$ and $BA=B$

Take, $AB=A$

$\Rightarrow A\left(BA\right)=A$ [ Since, $B=BA$ ]

$\Rightarrow \left(AB\right)A=A$

$\Rightarrow AA=A$ [ Since, $AB=A$ ]

$\Rightarrow A^2=A$

Therefore, $A$ is an idempotent matrix.

Take, $BA=B$

$\Rightarrow B\left(AB\right)=B$ [ Since, $A=AB$ ]

$\Rightarrow \left(BA\right)B=B$

$\Rightarrow BB=B$ [ Since, $BA=B$ ]

$\Rightarrow B^2=B$

Therefore, $B$ is an idempotent matrix.

Now, $A=AB$

Applying transpose on both sides, we get

$A^T=(AB{)}^T$

$\Rightarrow A^T=B^T.A^T....\left(i\right)$

Also, $B=BA$

Applying transpose on both sides, we get

$B^T=(BA{)}^T$

$\Rightarrow B^T=A^T.B^T...\left(ii\right)$

From equation $\left(i\right)$ and equation $\left(ii\right)$, we get

$(A^T{)}^2=A^T$ and

$(B^T{)}^2+B^T$

Therefore, $A^T$ and $B^T$ are alos idempotent matrices.

Hence, the correct options are $\left(A\right),\left(B\right)$ and $\left(C\right)$.