Question

If $A$ and $B$ are two matrices such that $AB=B$ and $BA=A,$ then

• A. $A$ is nilpotent
• B. $A+B$ is idempotent
• C. $A-B$ is idempotent
• D. $A-B$ is nilpotent

Answer 1

The correct option is D

$A-B$ is nilpotent

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

$\Rightarrow BAB=B^2$

$\Rightarrow \left(BA\right)B=B^2$

$\Rightarrow AB=B^2$

$\Rightarrow B=B^2$

Hence, $B$ is idempotent and similarly is $A$

$(A–B{)}^2=A^2–AB–BA+B^2$

$=A–B–A+B=0$

$\Rightarrow A–B$ is nilpotent.