Laplace Method for calculate the Determinant of cubic-matrix of order 2 and order 3

Abstract

In this paper, in continuation of our work, on the determinants of cubic -matrix of order 2 and order 3, we have analyzed the possibilities of developing the concept of determinant of cubic-matrix with three indexes, studying the possibility of their calculation according the Laplace expansion method's. We have noted that the concept of permutation expansion which is used for square determinants, as well as the concept of Laplace expansion method used for square and rectangular determinants, also can be utilized to be used for this new concept of 3D Determinants. In this paper we proved that the Laplace expansion method's is also valid for cubic-matrix of order 2 and order 3, these results are given clearly and with detailed proofs, they are also accompanied by illustrative examples. We also give an algorithmic presentation for the Laplace expansion method's.