We let $S=\mathbb C[x_{i,j}]$ denote the ring of polynomial functions on the space of $m\times n$ matrices and consider the action of the group $\mathrm{GL}=\mathrm{GL}_{m}\times\mathrm{GL}_{n}$ via row and column operations on the matrix entries. For a $\mathrm{GL}$-invariant ideal $I\subseteq S$, we show that the linear strands of its minimal free resolution translate via the BGG correspondence to modules over the general linear Lie superalgebra $\mathfrak{gl}(m|n)$. When $I=I_{\lambda}$ is the ideal generated by the $\mathrm{GL}$-orbit of a highest weight vector of weight $\lambda$, we give a conjectural description of the classes of these $\mathfrak{gl}(m|n)$-modules in the Grothendieck group, and prove that our prediction is correct for the first strand of the minimal free resolution.