Let $n \in \mathbb{N}$. Show that the determinant map $\det : M_{n \times n}(\mathbb{R}) \rightarrow \mathbb{R}$ is infinitely differentiable and compute the total derivative $d(\operatorname{det})$ at every point $A \in M_{n \times n}(\mathbb{R})$. Find a necessary and sufficient condition on the rank of $A$ for $d(\operatorname{det}) = 0$ at $A$.