Let $\omega$ be a complex number such that $\omega^3=1$ , but $\omega\ne1$. If: $A= \begin{pmatrix}1& \omega & \omega^2 \\ \omega& \omega^2&1\\\omega^2&\omega&1\end{pmatrix}$ then which of the following are true?
- $A$ is invertible.
- $\operatorname{rank}(A)=2$.
- $0$ is an eigenvalue of $A$.
- There exist linearly independent vectors $v,w\in\mathbb{C}^3$ such that $Av=Aw=0$ .
Statement 1 is false as rank is $2$. So 2 and 3 are true, but I have no idea about 4. Please help me.