(i) Let $G$ be the cyclic subgroup of $G L_{2}(\mathbf{C})$ generated by the matrix $\left(\begin{array}{cc}1 & 2 \\ 0 & -1\end{array}\right)$, acting on the polynomial ring $\mathbf{C}[X, Y]$. Determine the ring of invariants $\mathbf{C}[X, Y]^{G}$.

(ii) Determine $\mathbf{C}[X, Y]^{G}$ when $G$ is the cyclic group generated by $\left(\begin{array}{cc}0 & -1 \\ 1 & -1\end{array}\right)$.

[Hint: consider the eigenvectors.]