For a lattice point configuration A, Gelfand, Kapranov, and Zelevinsky introduced the principal A-determinant, a central notion in their theory of multidimensional discriminants. While their theory is controlled by the geometry of the associated toric variety, we consider the reciprocal linear space, whose geometry is controlled by the matroid of A. We propose a natural analogue of the principal A-determinant in this context and show that its geometric properties—such as projective duality, Chow form descriptions, and Horn-Kapranov parametrization—parallel the classical case. Furthermore, we develop the D-module counterpart: the reciprocal A-hypergeometric system. We prove it is holonomic, admits an integral representation, and has the Euler discriminant as its singular locus. This is ongoing joint work with Simon Telen (MPI MiS).