Q. If $A$ and $B$ are square matrices of order $n$, then $A - \lambda I$ and $B - \lambda I\$commute for every scalar $\lambda,$ only if
A) $AB = BA$
B) $AB + BA = 0$
C) $A = - B$
D) None of these
Explanation
(a)
$(A - \lambda I)(B - \lambda I) = (B - \lambda I)(A - \lambda I)$
$\Rightarrow AB - \lambda(A + B)I + \lambda^{2}I^{2} = BA - \lambda(B + A)I + \lambda^{2}I^{2}$
$\Rightarrow AB = BA$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.