Q. If $A$ is a nilpotent matrix of index 2, then for any positive integer$\ n,A(I + A)^{n}$ is equal to
A) $A^{- 1}$
B) $A$
C) $A^{n}$
D) $I_{n}$
Explanation
(b)
$A^{2} = O,\ A^{3} = A^{4} = \cdots = A^{n} = O$
Then, $A(I + A)^{n} = A(I + nA) = A + nA^{2} = A$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.