A combinatorial proof of Cramer's Rule

TL;DR

This paper provides a new, purely combinatorial and self-contained proof of Cramer's Rule, a classical method for solving linear systems, originally stated by Gabriel Cramer in 1750.

Contribution

It introduces a novel combinatorial proof of Cramer's Rule, offering a fresh perspective and a self-contained approach to this classical linear algebra result.

Findings

Provides a combinatorial proof of Cramer's Rule

Enhances understanding of linear algebra solutions

Offers a self-contained proof accessible to broader audiences

Abstract

In 1750, Gabriel Cramer famously stated, without proof, his eponymous rule for solving a system of linear equations with as many equations as unknowns. We give a purely combinatorial, and purely self-contained, proof of this old chestnut, still useful after all these years.