Linear Algebra – Determinant of Adjugate Matrix

Question

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,

Answer 1

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} $