Small Matrices with Small Inverses: Unimodular Zerofree Cases

Abstract

We consider unimodular matrices $M$ such that neither $M$ nor $M^{-1}$ contain zero entries. Matrices typically exhibit a trade-off: small $M$ imply large $M^{-1}$. We investigate rare cases where both remain small, classify these matrices up to symmetry, and discuss aspects of this balanced setting.

13 pages; proof of 2x2 counting formula (Euler totient-based) and more empirical results in Supplement