# On the probability of generating matrix incidence rings

## TL;DR

This paper calculates the probability that randomly chosen matrices and scalars generate a finite incidence ring, proving that all finite-dimensional incidence algebras over real and complex fields are generated by two matrices.

## Contribution

It provides a probabilistic analysis of matrix generation of incidence rings and establishes that two matrices suffice to generate all such algebras over real and complex fields.

## Key findings

- Probability of generating finite incidence rings is computed.
- All finite-dimensional incidence algebras over reals and complexes are generated by two matrices.

## Abstract

The probability that a tuple of matrices together with all scalars generates a finite incidence ring is calculated. It is proved that all real and complex finite-dimensional incidence algebras are generated by two randomly chosen matrices.