Small Matrices with Large Inverses: Unimodular $4 \times 4$ Cases

Steven Finch

combinatorics

Small Matrices with Large Inverses: Unimodular $4 \times 4$ Cases

arXiv:2607.07688

summary

The paper investigates how close a 4×4 unimodular matrix can be to singularity, analyzing ternary and non‑negative cases as the entry range expands.

Abstract

How close to singularity can an $n \times n$ unimodular matrix be? For ternary cases as $n$ increases, exact expressions are unlikely, but upon fixing $n = 4$ and assessing $(2 k + 1)$ -ary cases as $k$ increases, we make significant progress; similarly for $(k + 1)$ -ary cases of $4 \times 4$ nonnegative unimodular matrices.

21 pages; $5 \times 5$ conjectures appear at end; $6 \times 6$ and $7 \times 7$ signed conjectures likewise

Topics & keywords

#unimodular matrices #matrix singularity #integer matrices #nonnegative matrices #combinatorial matrix theoryunimodular4x4determinantternary matricesnonnegative matricesinteger entries

Small Matrices with Large Inverses: Unimodular 4×4 Cases

Abstract

Topics & keywords

Small Matrices with Large Inverses: Unimodular $4 \times 4$ Cases