Let $A\in$M$_n(K)$ with $K$ a field be of rank $n-1$. Then its adjugate matrix $A^{adj}$ has rank 1, as can be seen from $A^{adj}A=\det(A)I_n=0$, since the space of columns of $A$ is the kernel of $A^{adj}$ (some cofactor is nonzero).
This implies that any two columns of $A^{adj}$ are linearly dependent. Since the adjugate is polynomially computed from the original matrix, and rank $n-1$ implies the polynomial equation $\det(A)=0$, my question is: can we use this last equation (or any other) to find a explicit linear dependence relation between columns of the adjugate (e.g. by Laplace expansions or similar techniques)? I mean a relation in a generic sense, valid for all matrices of rank $n-1$.
Note that if this can be done, then it can be done for matrices over any commutative ring.
