Year: 2026
Type: preprint
Abstract: In this article, we study the invertibility of matrices of the form $A+EDF$, where $A$ and $D$ are square matrices over a unital, not necessarily commutative, ring and $A$ is singular. Under natural hypotheses on kernels and images, we prove that $A+EDF$ is invertible if and only if $D$ is invertible, and obtain an explicit formula for the inverse of $A+EDF$ by purely algebraic methods. When the coefficient ring is stably finite, or more specifically a field, we obtain an invertibility criterion under weaker hypotheses. Our results extend and sharpen a theorem of Eriksson and Nordqvist for complex matrices.
Source: arXiv (Cornell University)
Author: Öinert, Johan
Language: English
Topic: Rings, Modules, and Algebras
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7216457257