Consider $S = \{ A \in M_{n}(\mathbb{R}) : 0 < \operatorname{rank}(A) = p < n\}$, where p is a fixed integer.
Suppose that $n = 2$ and $p = 1$. Show that, locally, $S$ is the graph of a real $C^1$ function. Find the tangent plane of $S$ in $$A_{0} = \pmatrix{ 0 & 0\\ 0 & 1 }$$
I don't have an interesting idea to prove this, maybe use the rank theorem, but I don't know how can I use it. Someone can help me? Thank you in advance