The Fermat-Type Matrix Equation over $\mathrm{GL}_2(\mathbb{Z})$

Abstract

We determine the complete ordered solution set of the Fermat-type matrix equation $X^n+Y^n=Z^n$ in $\mathrm{GL}_2(\mathbb{Z})$ for every integer $n\ge3$, expressed in terms of simultaneous integral-conjugacy orbits while keeping the variable symmetries of the equation separate. The equation is solvable if and only if $4\nmid n$ and $6\nmid n$. Thus, in contrast with the determinant-one setting, odd multiples of $3$ are solvable in the full group $\mathrm{GL}_2(\mathbb{Z})$ and necessarily involve determinant-$-1$ components. For every admissible even exponent, the full ordered solution set has a canonical disjoint orbit decomposition indexed by the same transversal that occurs in the matrix Pythagorean equation. For odd exponents with $3\nmid n$, the commuting solutions form exactly six simultaneous-conjugacy orbits. The noncommuting odd solutions form exactly twenty simultaneous-conjugacy orbits when $3\nmid n$, and exactly fourteen when $3\mid n$. Under the coarser equivalence generated by signed variable permutations and simultaneous integral conjugation, these noncommuting orbits collapse to four and three classes, respectively. The proof combines the canonical orbit classification of the matrix Pythagorean equation, Cayley--Hamilton recurrences, commutator divisibility, trace-growth estimates, integral conjugacy of torsion matrices, and an elementary symmetry-reduced trace argument for the cubic equation.