A double Sylvester determinant

Abstract

Given two (n+1) × (n+1)-matrices A and B over a commutative ring, and some k ∈ {0, 1, …, n}, we consider the (n choose k) × (n choose k)-matrix W whose entries are (k+1) × (k+1)-minors of A multiplied by corresponding (k+1) × (k+1)-minors of B. Here we require the minors to use the last row and the last column (which is why we obtain an (n choose k) × (n choose k)-matrix, not a (n + 1 choose k + 1) × (n + 1 choose k + 1)-matrix). We prove that the determinant det W is a multiple of det A if the (n+1, n+1)-th entry of B is 0. Furthermore, if the (n+1, n+1)-th entries of both A and B are 0, then det W is a multiple of (detA)(detB). This extends a previous result of Olver and the author.