2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/220936We show that for all $k\ge 1$, there exists an integer $N(k)$ such that for all $n\ge N(k)$ the $k$-th order jet scheme over the commuting $n\times n$ matrix pairs scheme is reducible. At the other end of the spectrum, it is known that for all $k\ge 1$, the $k$-th order jet scheme over the commuting $2\times 2$ matrices is irreducible: we show that the same holds for $n=3$.To appear in Proceedings of the AMSAlgebraic GeometryRings and Algebras15A30text