Science:Math Exam Resources/Courses/MATH152/April 2016/Question B 04 (c)
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Question B 04 (c) 

Suppose in the year 2020, 50 million people live in cities and 50 million in the suburbs. Every year, 10% of city residents move to the suburbs and 20% of the residents of the suburbs move to cities. (c) Find the eigenvalues of P and a basis of eigenvectors. 
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Hint 

We should find eigenvalues and eigenvectors of matrix found in part (a). 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. For , we solve to find the eigenvalues. So, we have We then get that and We now find the corresponding eigenvector to the eigenvalue : This implies that and therefore Now letting we have and Let be an eigenvector corresponding to the eigenvalue . Similarly, we have We then get that and therefore Now letting we have and Therefore, we have two eigenvalues and the basis of eigenvector as 
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