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A complex matrix $A$ is ray-nonsingular if det$(X\circ {A})\neq
{0}$ for every matrix $X$ with positive entries. It is known taht
the order of a full ray-nonsingular matrix is at most 5 and
examples of full $n\times n$ ray-nonsingular matrices for
$n=2,3,4$ exist. In this note, we describe a property of a special
full $5\times5$ 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 $5\times5$
ray-pattern matrix that satisfies all three of the properties
given by Lee et al.
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