Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. Using the result $\exp(\mathbb{C}[A]) = (\mathbb{C}[A])^*$, conclude that $\exp(\mathscr{M}_n(\mathbb{C})) = \mathrm{GL}_n(\mathbb{C})$.