(A) Find the limiting matrix P for the standard form P̅ found in Example 3.

(B) Use P̅ to find the limit of the successive state matrices for S₀ = [0 0 1].

(C) Use P̅ to find the limit of the successive state matrices for S₀ = [.5 0 .5].

Data from Example 3

Two competing real estate companies are trying to buy all the farms in a particular area for future housing development. Each year, 20% of the farmers decide to sell to company A, 30% decide to sell to company B, and the rest continue to farm their land. Neither company ever sells any of the farms they purchase.

Draw a transition diagram and determine whether or not the Markov chain is absorbing.

Write a transition matrix that is in standard form.

If neither company owns any farms at the beginning of this competitive buying process, estimate the percentage of farms that each company will purchase in the long run.

If company A buys 50% of the farms before company B enters the competitive buying process, estimate the percentage of farms that each company will purchase in the long run.