A New Determinantal Formula for Three Matrices

Abstract

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

6 pages