Science:Math Exam Resources/Courses/MATH307/April 2012/Question 08 (b)
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Question 08 (b) 

Consider the stochastic matrix (b) By considering P^{2}, what can you say about the other eigenvalues of P? Give a reason. 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Science:Math Exam Resources/Courses/MATH307/April 2012/Question 08 (b)/Hint 1 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. We will begin by calculating what is:
We know that one of the properties of a stochastic matrix is, if or some power has all positive entries (that is, no zero entries) then the other eigenvalues of besides will have (referenced from page 187 of the current 2014 Summer Math 307 Textbook). Since we just found that all the entries in are positive, we can see that the other eigenvalues of must follow: . 