For fixed positive integers n, m, let \(\textrm{Mat}_{n\times m}(\mathbb {C})\) be the affine space consisting of all \(n\times m\) complex matrices, and let \(\mathbb {C}[\textbf{x}_{n\times m}]\) be its coordinate ring. For \(0\le r\le \min \{m,n\}\) , we apply the orbit harmonics method to the finite matrix loci \(\mathcal {Z}_{n,m,r}\) of rook placements with exactly r rooks, yielding a graded \(\mathfrak {S}_n\times \mathfrak {S}_m\) -module \(R(\mathcal {Z}_{n,m,r})\) . We find one signed and two sign-free graded character formulae for \(R(\mathcal {Z}_{n,m,r})\) . We also exhibit some applications of these formulae, such as proving a concise presentation of \(R(\mathcal {Z}_{n,m,r})\) , and proving some module injections and isomorphisms. Our techniques for showing the presentation of \(R(\mathcal {Z}_{n,m,r})\) also apply to involution matrix loci.
