Q. Let $A$ be an $n^{th}$-order square matrix and $B$ be its adjoint, then$\left| AB + KI_{n} \right|$ is (where $K$ is a scalar quantity)
A) $\left( |A| + K \right)^{n - 2}$
B) $\left( |A| + K \right)^{n}$
C) $\left( |A| + K \right)^{n - 1}$
D) None of these
Explanation
(b)
We have, $AB = A(adjA) = |A|I_{n}$
$\therefore AB + KI_{n} = |A|I_{n} + KI_{n}$
$\Rightarrow \ AB + KI_{n} = (|A| + k{)I}_{n}$
$\Rightarrow \left| \ AB + KI_{n} \right| = \left| \left( |A| + k \right)I_{n} \right|\left( \because\left| {\alpha I}_{n} = \alpha^{n} \right| \right)$
$= \left( |A| + K \right)^{n}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.