The solution of the Binet-Cauchy functional equation for square matrices

AbstractIt is shown that if ƒ : Mn(K)→K is a nonconstant solution of the Binet-Cauchy functional equation ƒ(AB) = 1n! ∑|s|=n nsƒ(As)ƒ(Bs) for A, B ∈ Mn(K) and if ƒ(E) = 0 where E is the n × n matrix with all entries 1n then ƒ is given by ƒ(A) = m(det A) where m is a multiplicative function on K. For ƒ(E)≠0 it has been shown by Heuvers, Cummings and Bhaskara Rao, that ƒ(A) = φ(per A) where φ is an isomorphism of K. Thus the Binet-Cauchy functional equation is the source of the common properties of det A and per A. The value of ƒ(E) is sufficient to distinguish between the two functions