###### Exercise 2

Every year, people move between urban (U), suburban (S), and rural (R) populations with the probabilities given in FigureĀ 4.5.12.

Construct the stochastic matrix \(A\) describing the movement of people.

Explain what the Perron-Frobenius theorem tells us about the existence of a steady-state vector \(\qvec\) and the behavior of a Markov chain.

Use the Sage cell below to find the some terms of a Markov chain.

Describe the long-term distribution of people among urban, suburban, and rural populations.