ELA, Volume 11, pp. 212-240, October 2004, abstract.

Non-existence of 5 x 5 Full Ray-nonsingular Matrices

Chi-Kwong Li, Thomas Milligan, and Bryan L. Shader

An n x n complex matrix is full ray-nonsingular if 
it has no zero entries and every matrix obtained by
changing  the magnitudes of its entries is nonsingular.
It is shown that a 5 x5 full ray-nonsingular matrix 
does not exist. This, combined with earlier results, 
shows that there exists an n x n full ray-nonsingular 
matrix if and only if n <= 4.