Degrees of projections of rank loci

We provide formulas for the degrees of the projections of the locus of square matrices with given rank from linear spaces spanned by a choice of matrix entries. The motivation for these computations stem from applications to `matrix rigidity'; we also view them as an excellent source of examples to test methods in intersection theory, particularly computations of Segre classes. Our results are generally expressed in terms of intersection numbers in Grassmannians, which can be computed explicitly in many cases. We observe that, surprisingly (to us), these degrees appear to match the numbers of Kekul\'e structures of certain `benzenoid hydrocarbons', and arise in many other contexts with no apparent direct connection to the enumerative geometry of rank conditions.