Generalizations of Sylvester's determinantal identity

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.