Permanents of $2\times 2$ Matrices Modulo $n$

TL;DR

This paper investigates the enumeration of invertible 2x2 matrices over integers modulo n with a specified permanent value, expanding understanding of matrix properties in modular arithmetic.

Contribution

It provides a formula for counting invertible 2x2 matrices with a given permanent modulo n, a novel contribution to modular matrix theory.

Findings

Derived explicit counts for matrices with specified permanent values.

Extended the understanding of matrix invariants in modular settings.

Abstract

In this article we compute the number of invertible $2\times 2$ matrices with integer entries modulo $n$ whose permanents are congruent modulo $n$ to a given integer $x$.