Calculation of Two Classes of Determinants by Reduction Method and Order-Increase Method

This paper explores two systematic approaches for computing determinants of structured matrices using Laplace Expansion. The reduction (recursion) method leverages recursive expansion to decompose high-order determinants into lower-order counterparts, exploiting structural repetition. This method simplifies complex calculations by iteratively applying Laplace Expansion. The order-increase (edge) method strategically augments matrices with auxiliary rows and columns to transform them into solvable forms. Examples include converting a -order determinant into an upper triangular matrix and extending a -order matrix into a Vandermonde determinant, enabling direct evaluation via established formulas. Both methods highlight how Laplace Expansion, combined with matrix structure insights, streamlines determinant computation. The reduction method is ideal for matrices with recursive patterns, while the edge method benefits determinants missing key rows/columns but amenable to structural augmentation. Practical applications span linear algebra, physics, and cryptography, where efficient determinant evaluation is critical. The paper underscores the pedagogical and computational value of these techniques, offering educators and researchers accessible strategies for tackling high-order determinants. Future directions include integrating these methods with computational tools and exploring broader interdisciplinary applications.