Matrix and eigenvectors 
$\quad$The matrix $\mathbf A=\frac19\begin{bmatrix}
7   & -2  &  0 \\
-2  & 6   &  3  \\
0   & 2   &  5   \\
\end{bmatrix}$ has eigenvalues $1$, $\frac23$ and $\frac13$n with the corresponding eigenvectors $ \mathbf v_1=\begin{bmatrix}
-2                  \\
2                    \\
1                     \\
\end{bmatrix}$, $\mathbf{v}_2=\begin{bmatrix}
2                       \\
1                        \\
2                         \\
\end{bmatrix}$, $\mathbf v_3=\begin{bmatrix}
1                           \\
2                            \\
-2                            \\
\end{bmatrix}$.                
$\text{(i)}$ Find the general solution of the equation $\mathbf x_{k+1}=\mathbf{Ax}_k$ if $\mathbf x_0=\begin{bmatrix}
1                                  \\
11                                  \\
-2                                   \\
\end{bmatrix}$.

For this question, does it mean I should find out what $x_k$ is in terms of $A$? 
Can anyone help me please?

Here is a HINT: $$x_k = A^k x_0$$
