Some 3D-Determinant Properties for Calculating of Cubic-Matrix of Order 2 and Order 3

Abstract

In this paper, as a continuation of our work on the determinant of cubic matrices, we have studied some properties of calculation of the determinant of cubic matrices for the cubic matrix of the order 2×2, as well as the order 3×3. Like properties of square determinants, these properties are analogous to some properties for determinants of the square matrix we have proved and noted that these properties also are applicable (or not in some details) to this concept for the determinant of the cubic matrix of orders 2×2 and order 3×3. All properties that have been proved in this paper, are proved for each case for orders 2×2 and order 3×3, by including all elements of the matrix, as well as all horizontal plans, horizontal layers, and vertical layers. All results in this paper, are presented in detail during the theorem proofs.

Keywords

• Cubic-matrix
• Determinant of cubic-matrix
• Determinant properties

References

1. Amiri, A., Fathy, M., & Bayat, M. (2010). Generalization of some determinantal identities for non-square matrices based on Radic’s definition. TWMS J. Pure Appl. Math, 1(2), 163-175.
2. Artin, M. (1991). Algebra. Upper Saddle River: Prentice Hall.
3. Bretscher, O. (2005) Linear algebra with applications (3rd ed.). Upper Saddle River: Prentice Hall.