Question

If $A$ and $B$ are two matrices such that $AB=B,BA=A$ then $[A$ and $B$ are not null matrices].

• A. $A^2=A$
• B. $B^2=0$
• C. $A-B$ is idempotent
• D. $A-B$ is nilpotent

Answer 1

The correct option is A $A^2=A$

$AB=B$ $\&$ $BA=A$

$\Rightarrow AB=B$

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

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

$\Rightarrow B.B=B$

$\Rightarrow B^2=B$

Similarly, $A^2=A$

Hence, the answer is $A^2=A.$