Elementary Transformation and its Applications for Hybrid number matrices

Abstract

In this paper, we define the elementary transformations of hybrid number matrices and establish their upper triangularization process based on these transformations. We introduce a new determinant by means of a real representation matrix and present the sufficient conditions for the existence of LU decompositions, together with a counterexample showing that hybrid number matrices do not necessarily admit an LU decomposition. Furthermore, we investigate the inverse of the hybrid number matrix and provide both the existence condition and a concrete computation method.