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 the above

Answer 1

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

$A^T$ is idempotent.

Explanation for the correct options:

Option $A$: $A$ is idempotent.

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

$$\begin{gathered} AB=A \\ \Rightarrow A\left(BA\right)=A \\ \Rightarrow \left(AB\right)A=A \\ \Rightarrow \left(A\right)A=A \\ \Rightarrow A^2=A \end{gathered}$$

Hence, $A$ is idempotent.

Option $B$: $B$ is idempotent.

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

$$\begin{gathered} BA=B \\ \Rightarrow B\left(AB\right)=B \\ \Rightarrow \left(BA\right)B=B \\ \Rightarrow \left(B\right)B=B \\ \Rightarrow B^2=B \end{gathered}$$

Hence, $B$ is idempotent.

Option $C$: ${\mathrm{A}}^{\mathrm{T}}$ is idempotent.

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

$$\begin{gathered} AB=A \\ \Rightarrow {\left(AB\right)}^T=A^T \end{gathered}$$

$\Rightarrow B^T{A}^T=A^T$……………… $\left(1\right)$

$$\begin{gathered} BA=B \\ \Rightarrow {\left(BA\right)}^T=B^T \end{gathered}$$

$\Rightarrow A^T{B}^T=B^T$……………… $\left(2\right)$

Multiply ${\mathrm{A}}^{\mathrm{T}}$ both side,

$\Rightarrow \left(A^T{B}^T\right)A^T=B^T{A}^T$

$\Rightarrow A^T{B}^T{A}^T=B^T{A}^T$ [From equation $\left(1\right)$ $B^T{A}^T=A^T$ ]

$\Rightarrow A^T{A}^T=A^T$

$\Rightarrow {\left(A^T\right)}^2=A^T$

Hence, $A^T$ is idempotent.

Hence, options $A,B,C$ are the correct options.