Question

If $A$ is an invertible matrix of order $n$, then the determinant of $\text{adj}A$ is equal to:

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

Answer 1

The correct option is C ${\left|A\right|}^{n-1}$

$A\left(adjA\right)=|A|\left(I\right)$

$|A\left(adjA\right)|=|\left(|A|I\right)|$

$|A||adjA|=|A{|}^n\times |I|$

$|A||adjA|=|A{|}^n$

case 1: if $|A|\ne 0$

Then we get ,$|adjA|=|A{|}^{n-1}$

case 2: if $|A|=0$

Then,$|adjA|=0$

And, we again get $|adjA|=|A{|}^{n-1}$