A Mِbius Identity Arising from Modularity in a Matroid Bilinear Form

The matrix for the bilinear form of the flag space of a matroid has (with respect to an appropriate basis) a tensor product structure when the matroid has a modular flat K. When determinants are taken, an identity is obtained for the rho function (a certain product of the Mِbius and beta functions) summed over flats with a fixed intersection with K. When the identity is interpreted for Dowling lattices and finite projective spaces, identities with similar combinatorial proofs are obtained for binomial and Gaussian coefficients, respectively.