Determinantal variety

Remarks

A matrix has rank at most $r$ if and only if all its $(r + 1) \times (r + 1)$ minors vanish, so the two descriptions agree. For $k$ algebraically closed, $M_r$ is irreducible of dimension $r(m + n - r)$, and the ideal $I_{r+1}$ is prime. Since $M_r$ is defined by homogeneous equations, it also defines a projective variety in $\mathbb{P}^{mn - 1}$.

Example: $m = n = 2$ and $r = 1$. $M_1$ is the quadric cone $x_{11} x_{22} - x_{12} x_{21} = 0$ in $\mathbb{A}^4$, of dimension $3 = 1 \cdot (2 + 2 - 1)$.

Non-example: the set of $2 \times 2$ matrices of rank exactly $1$ is not a determinantal variety. It is not closed, since its closure contains the zero matrix, which has rank $0$.