A Permutation-Based Determinant for Cubic Matrices and Its Laplace-Type Expansions

This paper included, the generalized-concept of determinant for cubic-matrix. In this paper we define the determinant for cubic-matrix of order $n, \forall n\in\mathbb{N}$. Several properties have been studied and prove, which come as a generalization of the properties studied by us for cubic-matrix of order 2 and 3.
In this paper, we introduce a permutation-based extension of the classical determinant to cubic matrices of order $n \times n \times n$. 
The proposed construction is defined through a double permutation expansion over the symmetric group $S_n$, which naturally extends the classical Leibniz formula for determinants of square matrices. We establish a consistent algebraic framework for this determinant, denoted by $\det(A_{n\times n\times n})$, and develop its fundamental structural properties. In particular, we prove a complete set of Laplace-type expansion theorems with respect to horizontal layers, vertical pages, and vertical layers. These results demonstrate that the determinant is well-defined and invariant under different expansion directions. Furthermore, we show that the proposed determinant satisfies key algebraic properties analogous to the classical case, including alternating behavior under interchange of layers, vanishing under degeneracy conditions, and multilinearity with respect to each structural component of the cubic matrix. Finally, we discuss the recursive nature of the proposed formulation and its implications for algorithmic computation, showing that the determinant can be reduced to lower-order cubic determinants through a structured cofactor expansion. The results provide a foundation for further development of higher-dimensional determinant theory and its potential applications in multilinear algebra and computational mathematics.