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arXiv · 2308.04411

A New Determinantal Formula for Three Matrices

Dinesh Khurana·T. Y. Lam
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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 $\,S$.

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Dinesh Khurana, T. Y. Lam. 2023-08-08. A New Determinantal Formula for Three Matrices. https://arxiv.org/abs/2308.04411

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