Question

If $A$ is a square matrix of order $n$, then the value of $\left|adjA\right|$ is-

• A. ${\left|A\right|}^{n-1}$
• B. ${\left|A\right|}^n$
• C. ${\left|A\right|}^{n+1}$
• D. ${\left|A\right|}^{n+2}$

Answer 1

Answer: (A)

${\left|A\right|}^{n-1}$

$A^{-1}=\frac{adjA}{|A|}$
$\Rightarrow A\left(adjA\right)=|A|I$
applying determinant on both sides gives
$\Rightarrow |A\left(adjA\right)|=|A{|}^n$
$\Rightarrow |A||adjA|=|A{|}^n$
$\therefore |adjA|=|A{|}^{n-1}$
Hence, option A.