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To tackle this problem:

(i) Compute the stationary distribution for each stochastic matrix:
The stationary distribution \(\pi = (\pi_1, \pi_2)\) satisfies \(\pi P = \pi\) and \(\pi_1 + \pi_2 = 1\). We solve the system of linear equations for each matrix.

(ii) Interpret the stationary distribution for the long-run proportion living in the city (state A).

(iii) For the convergence time, simulate or compute using matrix powers to determine when the city population proportion comes within \(0.05\) of the stationary distribution.

Let me know if you'd like step-by-step computation for a specific matrix! Which matrix would you like me to start with?

For each 2 × 2 column stochastic matrix below, answer the following: (i) Compute the stationary distribution. (ii) Suppose state A represents living in the city, while state B represents living in the suburbs of a metropolitan area. In the long run, what proportion of the total population will live in the city? (iii) If 60% of the population live in the city in year 0, then how many years will it take for the city population to be within 0.05 of the proportion given by the stationary distribution?

Explore how to compute the stationary distribution for 2 × 2 column stochastic matrices, analyze long-run proportions for city and suburban populations, and calculate convergence times to stationary distributions. Suitable for advanced high school or college-level mathematics.

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