The inverse problems of the determinantal regions of ray pattern and complex sign pattern matrices

We study the inverse problems for the determinantal regions RA of the ray pattern matrices and the determinantal regions SA of the complex sign pattern matrices. We determine all, but eight possible exceptions (in two equivalent classes), the determinantal regions SA when A ranges over all complex square matrices. We also determine all the possible determinantal regions RA, except those regions which are the union of {0} and an open sector with an angle greater than π. We also answer several questions proposed in [J.-Y. Shao, H.-Y. Shan, The determinantal regions of complex sign pattern matrices and ray pattern matrices, Linear Algebra Appl. 395 (2005) 211–228] concerning the number of the connected components of the set RA {0} (and of the set SA {0}).