Q. If $A,\ B,\ A + I,\ A + B$ are idempotent matrices, then $AB$ is equal To
A) $BA$
B) $- BA$
C) $I$
D) O
Explanation
(b)
Given $A,\ B,\ A + I,\ A + B$ are idempotent. Hence,
$A^{2} = A,\ B^{2} = B,(A + I)^{2} = A + I\$and$(A + B)^{2} = A + B$
$\Rightarrow A^{2} + B^{2} + AB + BA = A + B$
$\Rightarrow A + B + AB + BA = A + B$
$\Rightarrow AB + BA = O$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.