Every year, people move between urban (U), suburban (S), and rural (R) populations with the probabilities given in FigureĀ 12.

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Figure4.5.12The flow between urban, suburban, and rural populations.

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

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

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

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