Population movement between two towns

Each year, \(10\%\) of the residents of town \(P\) move to town \(Q\), and \(20\%\) of the residents of town \(Q\) move to town \(P\). Everyone else stays. Assume nobody else enters or leaves the two towns.

This is modelled by \(\mathbf{T}=\begin{pmatrix} 0.9 & 0.2 \\ 0.1 & 0.8 \end{pmatrix}\), so that \(\mathbf{T}\begin{pmatrix} p \\ q \end{pmatrix}\) gives the populations one year after populations \(p\) and \(q\).

1. The populations are now \(5000\) in \(P\) and \(4000\) in \(Q\). Find the populations after one year and after two years.[3]
2. Find the populations for which \(\mathbf{T}\) leaves both populations unchanged, given that the total is \(9000\).[3]
3. Explain why the total population never changes under this model, and state one reason why the model may be unrealistic.[2]