It is by definition that entries of the adjugate matrix $\text{adj}(A)$ are the corresponding $(n-1)$-minors of $A$ (up to a sign). What can we say about the $k$-minor of $\text{adj}(A)$ in relation to minors of $A$?
I have tried some cases starting from the definition of determinant (just like the proof of the Laplace expansion), but so far no luck. But I guess it is some kind of complementary minor in $A$.
Never mind...I find this Jacobi's theorem from Prasolov's (yes...every time...) book Problems and theorems in linear algebra:
General case follows immediately from permutating the rows and columns.