Combinatorial Identity Rule for the Sum of Matrix Determinants

Introduction

This algorithm explores a unique property related to the sum of determinants of matrices, providing a novel perspective on their interplay. The primary theory posits that the direct sum of determinants of matrices $A$ and $B$ doesn't equate to the determinant of their sum $A + B$. However, an intriguing pattern emerges, introducing an additional factor denoted as $c(A, B)$.

Overview

1. Basic Principle: The algorithm starts by stating that $\det(A + B) \neq \det(A) + \det(B)$.

2. Captivating Pattern: Despite the inequality, a captivating pattern is observed: $\det(A + B) \equiv \det(A) + \det(B) + c(A, B)$. Refer to the PDF for detailed mathematical expressions.

3. Illustration with Example: Matrices $A$ and $B$ are considered to illustrate the pattern. The algorithm demonstrates the pattern and showcases specific terms involved. For the mathematical details, please refer to the accompanying PDF.

4. Verification: The algorithm verifies the equality, illustrating that the left-hand side (LHS) is equivalent to both the right-hand side (RHS) in various combinations.

Extension to Larger Matrices

The pattern is asserted to hold for matrices of any size. The algorithm transforms matrices into a rank-3 tensor $a^{(\lambda)}_{ij}$, and by considering all possible combinations of $\lambda$ values, it calculates the determinant of the sum of matrices.

Conclusion

In conclusion, this algorithm provides a fresh perspective on the relationship between determinants of matrices and their sum. It introduces a unique insight into combinational arrangements, applicable to square matrices of varying dimensions. For a detailed mathematical explanation, please refer to the accompanying PDF.