Laplace Method for Calculate the Determinant of Cubic-Matrix of Order 2 and Order 3

Abstract

In this paper, as a continuation of our work on the determinants of cubic matrices of order 2 and order 3, we have investigated the possibilities of developing the concept of determinants of cubic matrices with three indexes, as well as the possibility of calculating them using the Laplace expansion method. We have observed that the notion of permutation expansion, which is used for square determinants, and the concept of the Laplace expansion method, which is used for square and non-square (rectangular) determinants, may be applied to this novel concept of 3D determinants. In this research, we demonstrated that the Laplace expansion approach is also applicable to cubic matrices of the second and third orders. These results are presented simply and with extensive proof. The findings are also supported by illustrated cases. In addition, we provided an algorithmic explanation for the Laplace expansion approach applied to cubic matrices.