The determinant of adjugate matrix I have the following proof that I would like to be walked through because I'm not intuitively seeing what to do:
If $A$ is $n\times n$, prove $\det\left(\operatorname{adj}(A)\right) = \det(A)^{n-1}$.
I know the property of $A\operatorname{adj}(A) = \det(A)I$ is important but I don't know how to apply it to get an answer. Any help is much appreciated.
Thanks,

A(adj A) = |A|(I)
|A(adj A)| = |(|A| I)|
|A| |adj A| = $|A|^n  * |I|$
|A| |adj A| = $|A|^n $
case 1: if  |A|$\neq0$
Then we get ,|adj A| = $|A|^{n-1}  $
case 2: if  |A|$=0$
Then,|adj A|$=0$
And, we again get |adj A| = $|A|^{n-1}  $
