The Determinant of Cubic-Matrix of order 2 and order 3: Some basic Properties and Algorithms
Armend Salihu, Orgest Zaka
TL;DR
This work extends permutation-based determinants to 3D cubic-matrices of order 2 and 3 by defining a map $\det$ on the set $\mathcal{M}_n(F)$ of cubic-matrices. It provides explicit formulas for $n=2$ as $\det[A_{2\times2\times2}] = a_{111}a_{222} - a_{112}a_{221} - a_{121}a_{212} + a_{122}a_{211}$ and a long permutation-expansion for $n=3$, with concrete examples yielding $\det=2$ and $63$ in sample matrices. The paper proves key properties mirroring 2D determinants, including $\det(I_2)=\det(I_3)=1$, zero determinants for matrices with a zero plane, and linearity under scaling of a single slice, as well as sign changes under interchanging consecutive slices and plan-index interchanges. It also furnishes exact and optimized algorithms (pseudocode) for computing these determinants, enabling practical computation for order-2 and order-3 cubic-matrices and setting the stage for potential higher-dimensional generalizations. Overall, the work provides a computable, structurally rich extension of determinants to 3D matrices with clear mathematical properties and implementable algorithms.
Abstract
Based on geometric intuition, in this paper we are trying to give an idea and visualize the meaning of the determinants for the cubic-matrix. In this paper we have analyzed the possibilities of developing the concept of determinant of matrices with three indexed 3D Matrices. We define the concept of determinant for cubic-matrix of order 2 and order 3, study and prove some basic properties for calculations of determinants of cubic-matrix of order 2 and 3. Furthermore we have also tested several square determinant properties and noted that these properties also are applicable on this concept of 3D Determinants.