The Binet–Cauchy theorem for the hyperdeterminant of boundary format multi-dimensional matrices

Let A, B be multi-dimensional matrices of boundary format ∏i=0p(ki+1), ∏j=0q(lj+1), respectively. Assume that kp=l0 so that the convolution A∗B is defined. We prove that Det (A∗B)= Det (A) α · Det (B) β where α=l0!/(l1!…lq!), β=(k0+1)!/(k1!…kp−1!(kp+1)!), and Det is the hyperdeterminant. When A, B are square matrices, this formula is the usual Binet–Cauchy Theorem computing the determinant of the product A·B. It follows that A∗B is nondegenerate if and only if A and B are both nondegenerate. We show by a counterexample that the assumption of boundary format cannot be dropped.