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. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$ For $a \in \mathbb{R}$, we define the application $$\begin{array}{ccc} Z_a : [0,1] & \longrightarrow & \mathbb{C} \\ t & \longmapsto & t + iat(1-t) \end{array}$$ Using the result of question 10b, deduce that $(\mathbb{C}[A])^*$ is path-connected.
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. We denote
$$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$
For $a \in \mathbb{R}$, we define the application
$$\begin{array}{ccc} Z_a : [0,1] & \longrightarrow & \mathbb{C} \\ t & \longmapsto & t + iat(1-t) \end{array}$$
Using the result of question 10b, deduce that $(\mathbb{C}[A])^*$ is path-connected.