A New Determinantal Formula for Three Matrices

For any three $n\times n$ matrices $A,B,X$ over a commutative ring $S$, we prove that $\mathrm{d}\mathrm{e}\mathrm{t}(A+B-AXB)=\mathrm{d}\mathrm{e}\mathrm{t}(A+B-BXA)\in S$. This apparently new formula may be regarded as a ``ternary generalization’’ of Sylvester’s classical determinantal formula $\mathrm{d}\mathrm{e}\mathrm{t}(I_n-AB)=\mathrm{d}\mathrm{e}\mathrm{t}(I_n-BA)$ for any pair of $n\times n$ matrices $A,B$ over $S$.