  In previous work, the authors studied the compressed shift operators
$S_{z_1}$ and $S_{z_2}$ on two-variable model spaces $H^2(\mathbb{D}^2)\ominus
\theta H^2(\mathbb{D}^2)$, where $\theta$ is a two-variable scalar inner
function. Among other results, the authors used Agler decompositions to
characterize the ranks of the operators $[S_{z_j}, S^*_{z_j}]$ in terms of the
degree of rational $\theta.$ In this paper, we examine similar questions for
$H^2(\mathbb{D}^2)\ominus \Theta H^2(\mathbb{D}^2)$ when $\Theta$ is a
matrix-valued inner function. We extend several results our previous work
connecting $\text{Rank} [S_{z_j}, S^*_{z_j}]$ and the degree of $\Theta$ to the
matrix setting. When results do not clearly generalize, we conjecture what is
true and provide supporting examples.
