With $\mathcal{Q}_{q,n}$ the distribution of $n$ minus the rank of a matrix chosen uniformly from the collection of all $n\times(n+m)$ matrices over the finite field $\mathbb{F}_{q}$ of size $q\ge2$, and $\mathcal{Q}_{q}$ the distributional limit of $\mathcal{Q}_{q,n}$ as $n\rightarrow\infty$, we apply Stein’s method to prove the total variation bound

\[\frac{1}{8q^{n+m+1}}\leq\|\mathcal{Q}_{q,n}-\mathcal{Q}_{q}\|_{\mathrm{TV}}\leq\frac{3}{q^{n+m+1}}.\] In addition, we obtain similar sharp results for the rank distributions of symmetric, symmetric with zero diagonal, skew symmetric, skew centrosymmetric and Hermitian matrices.