The $2\times2$ Matrix Mortality Problem and Invertible Matrices

TL;DR

This paper proves that the Matrix Mortality Problem is decidable for finite sets of 2x2 matrices with at most one invertible matrix, simplifying previous understandings of matrix invertibility's role.

Contribution

It introduces a modified proof demonstrating decidability of the problem under specific invertibility constraints, expanding the theoretical framework.

Findings

Decidability established for sets with at most one invertible matrix

Invertibility of matrices is shown to be irrelevant for the problem

Extension of previous proofs to broader matrix sets

Abstract

By modifying the proof of a paper by O. Bournez and M. Branicky, we establish that the Matrix Mortality Problem is decidable with any finite set of $2\times2$ matrices which has at most one invertible matrix. The same modification also shows that the number of non-invertible matrices is irrelevant.