  We 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 $n=3$.
