The sum of nonsingular matrices is often nonsingular

TL;DR

This paper investigates conditions under which the sum of two nonsingular matrices remains nonsingular, providing a general theorem and an application related to matrix sums.

Contribution

It presents a general theorem on nonsingular sums of matrices and explores their properties, extending previous understanding in matrix theory.

Findings

Many pairs of nonsingular matrices have nonsingular sums.

A general statement about nonsingular sums is proved.

An application demonstrating the theorem's utility is provided.

Abstract

If $M$ is a set of nonsingular $k\times k$ matrices then for many pairs of matrices, $A,B\in M,$ the sum is nonsingular, $\det(A+B)\neq 0.$ We prove a more general statement on nonsingular sums with an application.