Set $M_k:=V(I_k) \subset \operatorname{Mat}(m \times n) \simeq \mathbb{A}^{mn}$.

**(1)** If $k < \min\{m, \, n\}$ then  $M_{k-1}$ is precisely the singular locus of $M_k$. See 

E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris: *Geometry of Algebraic Curves*, p. 69.

**(2)** The determinantal variety $M_k$ is the $k$th secant variety to $M_1$, see MSE question [4384405][1].


  [1]: https://math.stackexchange.com/questions/4384405/on-the-equations-of-the-k-secant-of-the-segre-variety