Let $X=X_{n, m, r}$ be the set of $n \times m$ matrices of rank at most $r$ over a field $k$. Show that $X_{n, m, r}$ is naturally an affine subvariety of $\mathbf{A}^{n m}$ and that $X_{n, m, r}$ is a Zariski closed subvariety of $X_{n, m, r+1}$.

Show that if $r<\min (n, m)$, then 0 is a singular point of $X$.

Determine the dimension of $X_{5,2,1}$.