The Binet-Cauchy Theorem for the Hyperdeterminant of boundary format multidimensional Matrices

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Let A, B be multidimensional matrices of boundary format respectively ∏ᵢ₌₀ᵖ(kᵢ+1), ∏ⱼ₌₀q(lⱼ+1). 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.