the adjugate of the adjugate

Solution 1:

Exactly the same universal approach in the related problem below works on your similar problem.

Hint $\ $ Denote the adjoint of $\rm\:A\:$ by $\rm\:A^*.\:$ Then

$$\rm\: A A^* = |A|\: I_n \ \Rightarrow\ |A|\, |A^*| = |A|^n\ \Rightarrow\ |A^*| = |A|^{n-1}$$

where the cancellation of $\rm\:|A|\:$ is done universally, i.e. considering the matrix extries as indeterminates, so the determinant is a nonzero polynomial in a domain $\rm\:\mathbb Z[a_{ij}],$ hence is cancellable. For further discussion of such universal cancellation of "apparent singularities" see here and here and here.