On ray-nonsingular matrices

A complex matrix A is ray-nonsingular if det(X A)≠0 for every matrix X with positive entries. It is known that the order of a full ray-nonsingular matrix is at most 5 and examples of full n×n ray-nonsingular matrices for n=2, 3, 4 exist. In this note, we describe a property of a special full 5×5 ray-nonsingular matrix, if such matrix exists, using the concept of an isolated set of transversals and we obtain a necessary condition for a complex matrix A to be ray-nonsingular. Moreover we give an example of a full 5×5 ray-pattern matrix that satisfies all three of the properties given by Lee et al. [Discrete Math. 216 (2000) 221–233]. The notion of -ray nonsingularity is also introduced.