The determinant of adjugate matrix

Solution 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} $

Solution 2:

We have the relation $$A \operatorname{adj}(A)=\det(A)I $$ Now take determinant on both the sides we get, $$\det(A)\det(\operatorname{adj}(A))=\det(\det(A)I) \tag1$$ use the relation that, for a matrix $A$ that is $n \times n$,

$$\det(kA)=k^n (\det(A))$$ where k is a numerical constant. Hence we have

$$\det(\det(A)I)=\det(A)^n \det (I)=\det(A)^n \tag2$$ and from equations $(1)$ and $(2)$ it follows that $$\det(A)\det(\operatorname{adj}(A))=\det(A)^n ,$$ which implies $$\det(\operatorname{adj}(A)) = \det(A)^{n-1} .$$