Matrix completion problem involves determining whether or not a completion of a partial matrix exist for a certain class of matrices satisfying a number of prescribed properties or not. Research on completion of various classes of matrices, including P-matrices, P0-matrices, as well as Wss P0-matrices has been done. In particular, completion of Wss P0-matrixfor4×4 matrices have been explored using digraphs with 4 arcs. However, the case of digraphs of order 5 with up to 5 arcs has not been studied. In this study, therefore, the completions of non-isomorphic digraphs of order 5 with up to 5 arcs were determined. Digraphs were utilized to create partial Wss P0-matrix from which all principal minors were obtained. Partial matrices were extracted from non-isomorphic digraphs. Principal sub-matrices were extracted from each partial matrix thereby finding the determinant of each sub-matrix obtained. Zero completion was done to all partial matrices to ascertain the viability of completion for each partial matrix. Digraphs characteristics, which leads to completion or non-completion, were analyzed. These digraphs characteristics were derived from digraphs used to construct the partial Wss P0-matrix. This study established that all cyclic and acyclic digraphs of order 5 with up to 5 arcs were found to have zero completion into a Wss P0-matrix. Digraphs of order 5 with 2 arcs that have a positionally symmetric cycle were found to have completion. However, those digraphs of order 5 with 4 and 5 arcs that possess positionally symmetric cycles were discovered to have no completion. Insights gained from this class of matrix could be applied to fill gaps in data surveys, and business analytics, allocating resources, network modelling, and optimizing processes.