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

## Document Type

Article

## Publication Date

3-15-1991

## Department

Department of Mathematical Sciences

## Abstract

It is shown that if f{hook} : Mn(K)→K is a nonconstant solution of the Binet-Cauchy functional equation f{hook}(AB) = 1 n! ∑ |s|=n n sf{hook}(As)f{hook}(Bs) for A, B ∈ Mn(K) and if f{hook}(E) = 0 where E is the n × n matrix with all entries 1 n then f{hook} is given by f{hook}(A) = m(det A) where m is a multiplicative function on K. For f{hook}(E)≠0 it has been shown by Heuvers, Cummings and Bhaskara Rao, that f{hook}(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 f{hook}(E) is sufficient to distinguish between the two functions.

## Publication Title

Discrete Mathematics

## Recommended Citation

Heuvers, K.,
&
Moak, D.
(1991).
The solution of the Binet-Cauchy functional equation for square matrices.
*
Discrete Mathematics,
88*(1), 21-32.
http://doi.org/10.1016/0012-365X(91)90056-8

Retrieved from: https://digitalcommons.mtu.edu/michigantech-p/5241