Yet Another Proof of Sylvester's Determinant Identity

TL;DR

This paper presents a new proof of Sylvester's determinant identity using differential operators and properties of derivatives of determinants, enriching the collection of existing proofs.

Contribution

It introduces a novel proof method leveraging derivatives and the product rule, expanding the understanding of Sylvester's determinant identity.

Findings

New proof based on derivatives and the product rule

Complemented existing proofs with an elegant differential approach

Enhances the mathematical literature on determinant identities

Abstract

In 1857 Sylvester stated a result on determinants without proof that was recognized as important over the subsequent century. Thus it was a surprise to Akritas, Akritas and Malaschonok when they found only one English proof - given by Bareiss 111 years later! To rectify the gap in the literature these authors collected and translated six additional proofs: four from German and two from Russian. These proofs range from long and "readily understood by high school students" to elegant but high level. We add our own proof to this collection which exploits the product rule and the fact that taking a derivative of a determinant with respect to one of its elements yields its cofactor. A differential operator can then be used to replace one row with another.