Eigenvalues · Medium problem

Each year $80\%$ of city residents stay in the city and $20\%$ move to the suburbs, while $70\%$ of suburban residents stay and $30\%$ move to the city. This is modelled by $\mathbf{x}_{k+1} = A\mathbf{x}_k$ with $A = \begin{pmatrix}0.8 & 0.3\\0.2 & 0.7\end{pmatrix}$, where $\mathbf{x}_k$ lists the fractions (city, suburbs). Find the long-run fraction of the population living in the city, the first entry of the eigenvector for eigenvalue $1$ whose entries add to $1$.