Q. If $A$ and $B$ are two matrices such that $AB = B$ and $BA = A$, then
A) $\left( A^{5} - B^{5} \right)^{3} = A - B$
B) $\left( A^{5} - B^{5} \right)^{3} = A^{3} - B^{3}$
C) $A - B$ is idempotent
D) $A - B$ is nilpotent
Explanation
(b)
Since $AB = B\$and$\ BA = A,\$so
$BAB = B^{2}$
$\Rightarrow (BA)B = B^{2}$
$\Rightarrow AB = B^{2}$
$\Rightarrow B = B^{2}$
Hence, $B$ is idempotent and similarly$A$.
$(A - B)^{2} = A^{2} - AB - BA + B^{2} = A - B - A + B = 0$
Therefore, $A - B$ is nilpotent
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.