Rational solutions of certain matrix equations

A sign pattern is a matrix whose entries are from the set {+,-,0}. For a real matrix B, sgn(B) is the sign pattern obtained by replacing each positive (respectively, negative, zero) entry of B by + (respectively, -, 0). For a sign pattern A, the sign pattern class of A, denoted Q(A), is defined as {B:sgn(B)=A}.The purpose of this note is to prove the following theorem: Let A, B, and C be real matrices such that AB=C. Suppose that the zero bipartite graph of C (this is the same as the complement of the bipartite graph of C) is a forest. Then there exist rational perturbations and of A,B, and C, respectively, in the same corresponding sign pattern classes, such that .