Let $z$ be a generator of the cyclic group $\mathbb{Z}_3 = \{ 1,z,z^2 \}$. Prove that a representation $\rho$ of $\mathbb{Z}_3$ in the $2$-dimensional complex vector space $\mathbb{C}^2$ can be defined by $\rho(z) = \begin{bmatrix} -1 & 1 \\ -1 & 0 \end{bmatrix}.$
I know I need to check that $\rho : \mathbb{Z}_3 \to GL(\mathbb{C}^2)$ is a homomorphism, but how do I check that $\rho(g \circ h) = \rho(g) \circ \rho(h)$ for any $g,h \in \mathbb{Z}_3$?