A political consultant tells us that in his town the population is stable at 300,000 people. There are presently 150, 000 Independents, 90, 000 Democrats, and 60, 000 Republicans. Each year 20 percent of the Independents become Democrats and 10 percent become Republicans. Similarly, each year 20 percent of the Democrats become Independents and 10 percent become Republicans. Finally, each year 10 percent of the Republicans become Democrats and 10 percent become Independents. Let x(t) = (x1(t), x2(t), x3(t)) be the number of (Independents,Democrats,Republicans) after t years.
(a) What matrix A has the property that x(t + 1) = Ax(t)?
(b) Find x(t). (Hint: the eigenvalues of A are 1, 0.5, and 0.7).
(c) In the long run, how will the population be distributed between Independents, Democrats, and Republicans?