Fixed $n\in\mathbb{N}$, describe the irreducible components, and prove that there are ${n+1\choose{2}}$ of them, of the following algebraic variety: $$V=\{A\in M_{2}(\mathbb{C})\mid A^n=\mathbb{I}\}$$
For $n=2$, the components are $\{\mathbb{I}\}$, $\{-\mathbb{I}\}$ and
$$ \left\{ \begin{pmatrix} a & b\\ c & d \end{pmatrix}: \ \ a=-d \ , \ \ a^2+bc=1 \ \right\}$$ For the other values of $n$, I don't have idea how to proceed