Checking an eigenvector and diagonalizability

Verify that $v=(1,1,1)$ is an eigenvector of the matrix

$$A=\left(\begin{array}{ccc}2& 1& -1\\ 0& 3& -1\\ 1& 1& 0\end{array}\right),$$

determine all the eigenvalues of $A$ and determine whether $A$ is diagonalizable.