determinant of the adjugate

You know that A adj(A) = det(A) I, so take determinants:
det(A) det(adj(A)) = (det(A))^n

(I used det(AB)=det(A)det(B) and det(cA)=c^ndet(A).)

So if A is invertible then det(adj(A)) = (det(A))^(n-1). Otherwise det(adj(A))=0. Hence in both cases det(adj(A)) = (det(A))^(n-1).

If you multiply each of the n rows by c, then you multiply the determinant by c n times.

A^-1 = 1/det(A) * adj(A)


Take determinant of both sides
det(A^-1) = det(1/det(A) * adj(A))


Simplify
det(A^-1) = 1/det(A) * det(adj(A))
det(A) * det(A^-1) = det(adj(A))

So as you can see, you just need to find det(A) in order to know det(adj(A))

Note that the last post in this thread before yours is 8 years old. I suspect they are no longer waiting for an answer...

As OP reads it, a single tear rolls down their cheek. "If this answer had come just a week earlier, the world could have been saved!" they whisper, as across the globe the fires begin.