Determinants for nxn matrices and the symmetric Newton formula in the 3x3 case

TL;DR

This paper surveys various generalizations of determinants for matrices over non-commutative rings, introduces a trace expression for the symmetric determinant, and presents a symmetric Newton formula specifically for 3x3 matrices.

Contribution

It provides a comprehensive survey of determinant generalizations and introduces a symmetric Newton formula for 3x3 matrices over arbitrary rings.

Findings

Trace expression for symmetric determinant derived

Symmetric Newton formula established for 3x3 matrices

Survey of determinant generalizations included

Abstract

One of the aims of this paper is to provide a short survey on the Z2-graded, the symmetric and the left (right) generalizations of the classical determinant theory for square matrices with entries in an arbitrary (possibly non-commutative) ring. This will put us in a position to give a motivation for our main results. We use the preadjoint matrix to exhibit a general trace expression for the symmetric determinant. The symmetric version of the classical Newton trace formula is also presented in the 3x3 case.