A diophantine problem concerning third order matrices

TL;DR

This paper investigates a special class of third order unimodular matrices with entries not equal to ±1, exploring their properties when entries are cubed and establishing related determinant conditions.

Contribution

It introduces the existence of third order unimodular matrices with entries excluding ±1 that maintain unimodularity after cubing, and constructs matrices with specific determinant properties.

Findings

Existence of third order unimodular matrices with entries not ±1 that remain unimodular after cubing

Construction of matrices with determinant k and cubed determinant k^3, avoiding entries ±1

Extension of Diophantine problems to matrix determinants and entries

Abstract

In this paper we find a third order unimodular matrix, none of whose entries is $1$ or $-1$, such that when each entry of the matrix is replaced by its cube, the resulting matrix is also unimodular. Further, we find third order square integer matrices $(a_{ij})$, none of the integers $a_{ij}$ being $1$ or $-1$, such that $\det{(a_{ij})}=k$ and $\det{(a_{ij}^3)}=k^3$, where $k$ is a nonzero integer.