A New Determinantal Formula for Three Matrices

TL;DR

This paper introduces a novel determinantal identity involving three matrices over a commutative ring, extending classical results and providing a new perspective on matrix determinant relations.

Contribution

It presents a new determinantal formula for three matrices, generalizing Sylvester's classical identity to a ternary case over a commutative ring.

Findings

Proves that det(A+B-AXB) equals det(A+B-BXA) for three matrices over a commutative ring.

Establishes a ternary generalization of Sylvester's determinant identity.

Provides a new algebraic tool for matrix analysis and identities.

Abstract

For any three $\,n\times n\,$ matrices $\,A,B,X\,$ over a commutative ring $\,S$, we prove that $\,{\rm det}\,(A+B-AXB)={\rm det}\,(A+B-BXA) \in S$. This apparently new formula may be regarded as a ``ternary generalization'' of Sylvester's classical determinantal formula $\,{\rm det}\,(I_n-AB)={\rm det}\,(I_n-BA)\,$ for any pair of $\,n\times n\,$ matrices $\,A,B\,$ over $\,S$.