Generalizations of Sylvester's determinantal identity

TL;DR

This paper reviews various generalizations of Sylvester's determinantal identity, unifies their formulations, and introduces a new generalization relating determinants of matrices with bordered matrices, extending the classical identity.

Contribution

It unifies existing generalizations of Sylvester's identity and proposes a novel generalization involving determinants of matrices with multiple added rows and columns.

Findings

Unified formulation of multiple Sylvester's identity generalizations

Introduction of a new generalization involving bordered matrices

Classical Sylvester's identity as a special case

Abstract

In this paper we deal with the noteworthy Sylvester's determinantal identity and some of its generalizations. We report the formulae due to Yakovlev, to Gasca, Lopez--Carmona, Ramirez, to Beckermann, Gasca, M\"uhlbach, and to Mulders in a unified formulation which allows to understand them better and to compare them. Then, we propose a different generalization of Sylvester's classical formula. This new generalization expresses the determinant of a matrix in relation with the determinant of the bordered matrices obtained adding more than one row and one column to the original matrix. Sylvester's identity is recovered as a particular case.