  The main goal of the paper is the discussion of a deeper interaction between
matrix theory over polynomial rings over a field and typical methods of
commutative algebra and related algebraic geometry. This is intended in the
sense of bringing numerical algebraic invariants into the picture of
determinantal ideals, with an emphasis on non-generic ones. In particular,
there is a strong focus on square sparse matrices and features of the dual
variety to a determinantal hypersurface. Though the overall goal is not
exhausted here, one provides several environments where the present treatment
has a degree of success.
