The city of Smithville maintains a constant population of 300,000 people from year to year. A political science study estimated that there were 150,000 Independents, 90,000 Democrats, and 60,000 Republicans in the town. It was also estimated that each year 20% of the Independents become Democrats and 10% become Republicans. Similarly, 20% of the Democrats become Independents and 10% become Republicans, while 10% of the Republicans become Democrats and 10% become Independents. Let x_0 = [150,000 90,000 60,000]^T and let X_1 be a vector representing the number of people in each group after 1 year. a) Determine the transition matrix that describes this situation.
b) Find the eigenvalues of A and factor A into a product XDX^-1, where D is diagonal.
c) Compute lim_n rightarrow infinity A^n x and explain what this means in terms of the towns political spectrum.