The Binet-Pexider functional equation for rectangular matrices

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If K is a field of characteristic 0 then the following is shown. If f, g, h: Mn(K) →K are non-constant solutions of the Binet-Pexider functional equation {Mathematical expression} for rectangular matrices A ∈ Mn ×(n + r) (K) and B ∈ M(n + r) ×n(K), and a fixed non-negative integer r then f(X) = ab d(X), g(X) = a d(X), and h(X) = b d(X) where a and b are arbitrary constants from K and d: Mn (K) →K is a non-constant solution of the Binet-Cauchy functional equation {Mathematical expression} for A ∈ Mn ×m (K) and B ∈ MM ×n (K) where n ≤ m ≤ n + r. The general non-constant solution to (2) has been shown by Heuvers, Cummings, and K. P. S. bhaskara Rao, to be d(X) = φ(per X), where φ is an isomorphism of K, provided that d(E) ≠ 0 for the n × n matrix E with all entries 1/n. Heuvers and Moak have shown that the general non-constant solution to (2) is given by d(x) = μ(det X), where μ is a non-constant multiplicative function on K if m = n and d(E) = 0. If n ≤ m ≤n + 1 and d(E) = 0 they have shown that it is given by d(X) = φ(det X), where φ is an isomorphism of K. © 1990 Birkhäuser Verlag.

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Aequationes Mathematicae