Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Transcribed Image Text:3. The city of Mawroke maintains a constant voter population of 300,000 people per year. A political science study estimated that there were 150,000 Independents, 90,000 Democrats, and 60,000 Republicans in the city. It was also estimated that each year 20% of the Independents become Democrats and 10% become Republicans. Similarly, 20% of Democrats become Independents and 10% become Republicans, while 10% of Republicans defect to the Democrats and 10% become Independents each year.
(a) Find the stochastic transition matrix \( A \) such that \( x^k = A x^{k-1} \), \( k = 1, 2, 3, \ldots \), where \( x^k \) is a vector representing the number of people in each group.
(b) Find the eigenvalues of \( A \) and corresponding eigenvectors.
(c) Factor \( A \) into a \( PDP^{-1} \), where \( D \) is diagonal.
(d) Which group will dominate in the long run? Justify your answer by computing \( \lim_{n \to \infty} A^n x \).
(Note: Above all, a blue stain or mark is present on the right side of the text.)
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