Question

Let $A$ be a $n^{th}$ order square matrix and matrix $B$ be its adjoint, then $|AB+K{I}_n|$ is equal to (where $K$ is a scalar quantity)

• A. $(|A|+K{)}^{-1}$
• B. $(|A|+K{)}^{n-2}$
• C. $(|A|+K{)}^{n-1}$
• D. $(|A|+K{)}^n$

Answer 1

The correct option is D $(|A|+K{)}^n$

We have, $AB=A\left(adjA\right)=|A|I_n$

$\therefore AB+K{I}_n=|A|I_n+K{I}_n$

$\Rightarrow AB+K{I}_n=(|A|+K)I_n$

$\Rightarrow |AB+K{I}_n|=|(|A|+K)I_n|(\because |a{I}_n|=a^n)$

$\Rightarrow |AB+K{I}_n|=(|A|+K{)}^n$