### A double Sylvester determinant

#### Abstract

Given two (*n*+1) × (*n*+1)-matrices *A* and *B* over a commutative ring, and some *k* ∈ {0, 1, …, *n*}, we consider the (*n* choose *k*) × (*n* choose *k*)-matrix *W* whose entries are (*k*+1) × (*k*+1)-minors of *A* multiplied by corresponding (*k*+1) × (*k*+1)-minors of *B*. Here we require the minors to use the last row and the last column (which is why we obtain an (*n* choose *k*) × (*n* choose *k*)-matrix, not a (*n* + 1 choose *k* + 1) × (*n* + 1 choose *k* + 1)-matrix). We prove that the determinant det *W* is a multiple of det *A* if the (*n*+1, *n*+1)-th entry of *B* is 0. Furthermore, if the (*n*+1, *n*+1)-th entries of both *A* and *B* are 0, then det *W* is a multiple of (det*A*)(det*B*). This extends a previous result of Olver and the author.

#### Keywords

#### Full Text:

MANUSCRIPTDOI: https://doi.org/10.26493/1855-3974.2248.d3f

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications