A Generalized Determinant of Matrices and Applications

TL;DR

This paper introduces a generalized determinant concept that extends the traditional determinant to broader matrix classes, preserving key properties and enabling new applications like generalized Cramer's rule and volume calculations.

Contribution

It proposes a new generalized determinant definition compatible with classical properties and applicable to non-square matrices, expanding theoretical and practical matrix analysis tools.

Findings

Maintains multilinearity and alternation in the generalized determinant

Derives a generalized Cramer's rule for solving linear systems

Introduces a generalized oriented volume concept

Abstract

A generalized definition of the determinant of matrices is given, which is compatible with the usual determinant for square matrices and keeps many important properties, such as being an alternating multilinear function, keeping multiplication formula and partly keeping the Cauchy-Binet's formula. As applications of the new theory, the generalized Cramer's rule and the generalized oriented volume are obtained.