Laplace Method for calculate the Determinant of cubic-matrix of order 2 and order 3
Orgest Zaka, Armend Salihu
TL;DR
The paper addresses defining and computing determinants for cubic-matrices (3D arrays) of order $2$ and $3$ and extends the Laplace expansion technique to the 3D setting. It develops a determinant concept derived from permutation-like expansion and proves that Laplace expansion along any horizontal layer, vertical page, or vertical layer is valid for cubic-matrices, providing rigorous proofs and illustrative examples. Minors and cofactors for cubic-matrices are defined, enabling Laplace expansions and yielding invariant results across expansion directions; the work also supplies an algorithmic pseudocode for computing these determinants. The concrete formulations for order $2$ and $3$ and the accompanying numeric examples (e.g., $\,\det[A_{2\times2\times2}] = a_{111} a_{222} - a_{112} a_{221} - a_{121} a_{212} + a_{122} a_{211}$ and $\,\det[A_{3\times3\times3}]$ as an explicit alternating sum) support practical computation and potential software implementations, with a broader impact on the theory and application of 3D determinants in geometry and algebra.
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.