Relating the spectrum of a matrix and a principal submatrix using adjugates and Schur complements

Let $\mathcal{M}$ be a square matrix over a commutative ring and let $\mathcal{A}$ be a principal submatrix. We give relations between the determinants of $\mathcal{M}$ and $\mathcal{A}$ based on an annihilating polynomial for one of them. The intended application is the size reduction of complex latent root problems, especially the reduction of ordinary eigenvalue problems if a matrix or its principal submatrix have a low degree minimal polynomial. An example is the spectrum of vertex perturbed strongly regular graphs.