  This article generalizes joint work of the first author and I. Swanson to the
$s$-multiplicity recently introduced by the second author. For $k$ a field and
$X = [ x_{i,j}]$ a $m \times n$-matrix of variables, we utilize Gr\"obner bases
to give a closed form the length $\lambda( k[X] / (I_2(X) + \mathfrak{m}^{
\lceil sq \rceil} + \mathfrak{m}^{[q]} ))$ where $s \in \mathbf{Z}[p^{-1}]$,
$q$ is a sufficiently large power of $p$, and $\mathfrak{m}$ is the homogeneous
maximal ideal of $k[X]$. This shows this length is always eventually a {\it
polynomial} function of $q$ for all $s$.
