State what it means for a function $f: \mathbb{R}^{m} \rightarrow \mathbb{R}^{r}$ to be differentiable at a point $x \in \mathbb{R}^{m}$, and define its derivative $f^{\prime}(x) .$

Let $\mathcal{M}_{n}$ be the vector space of $n \times n$ real-valued matrices, and let $p: \mathcal{M}_{n} \rightarrow \mathcal{M}_{n}$ be given by $p(A)=A^{3}-3 A-I$. Show that $p$ is differentiable at any $A \in \mathcal{M}_{n}$, and calculate its derivative.

State the inverse function theorem for a function $f$. In the case when $f(0)=0$ and $f^{\prime}(0)=I$, prove the existence of a continuous local inverse function in a neighbourhood of 0 . [The rest of the proof of the inverse function theorem is not expected.]

Show that there exists a positive $\epsilon$ such that there is a continuously differentiable function $q: D_{\epsilon}(I) \rightarrow \mathcal{M}_{n}$ such that $p \circ q=\left.\mathrm{id}\right|_{D_{\epsilon}(I)}$. Is it possible to find a continuously differentiable inverse to $p$ on the whole of $\mathcal{M}_{n}$ ? Justify your answer.