Remarks on two theorems in linear algebra

TL;DR

This paper provides elementary proofs of the Cayley-Hamilton theorem and Birkhoff's theorem on bi-stochastic matrices using polynomial ring concepts, making these classical results more accessible.

Contribution

It introduces simplified, elementary proofs of two fundamental theorems in linear algebra using polynomial ring techniques.

Findings

Elementary proof of Cayley-Hamilton theorem

Elementary proof of Birkhoff's theorem on bi-stochastic matrices

Simplifies understanding of key linear algebra theorems

Abstract

In this note, we use the concept of a polynomial ring to give an elementary proof to Cayley-Hamilton Theorem. We also give an elementary proof to Birkhoff theorem on Bi-stochastic matrices.