A basis of an eigenspace

Kernel.
$$A-2I = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}.$$
The system is $y=0$ and $z=0$, so the kernel is $\{(x,0,0)\}$.

Conclusion. $E_2=\textrm{span}((1,0,0))$, dimension $1$.

Remark. The algebraic multiplicity of $\lambda=2$ is $2$ (it appears twice on the diagonal), but the geometric multiplicity is $1$, so $A$ is not diagonalizable.