A Short and Elementary Proof of the Two-sidedness of the Matrix-Inverse

TL;DR

This paper provides an elementary proof demonstrating that the inverse of a matrix is two-sided, emphasizing the role of linear independence and basis concepts without relying on elementary matrices.

Contribution

It introduces a new, elementary proof of the two-sidedness of matrix inverses using only basic linear algebra concepts, simplifying the understanding of invertibility.

Findings

Proof relies solely on linear independence and reduced row-echelon form.

Shows invertibility is equivalent to being row-equivalent to the identity matrix.

Provides a basis-centered proof of the invertible matrix theorem.

Abstract

An elementary proof of the two-sidedness of the matrix-inverse is given using only linear independence and the reduced row-echelon form of a matrix. In addition, it is shown that a matrix is invertible if and only if it is row-equivalent to the identity matrix without appealing to elementary matrices. This proof underscores the importance of a basis and provides a proof of the invertible matrix theorem.