Science:Math Exam Resources/Courses/MATH152/April 2012/Question 07 (b)
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Question 07 (b) 

Consider a random walk that has transition matrix: Find all eigenvalues of the transition matrix P (no need to calculate the eigenvectors at this stage). 
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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

As it always has, the eigenvalues of a matrix P are the zeros of the characteristic polynomial 
Hint 2 

For this simple matrix however, the eigenvalue beg to be spotted without any calculation.

Hint 3 

This should already reveal all eigenvalues of P. If you miss one hint, calculate the missing eigenvalue from the fact that the sum of the diagonal entries of a matrix equals the sum of the eigenvalues. 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 1 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. The eigenvalues of P satisfy Performing the calculation of the determinant using the cofactor method gives Solving gives the eigenvalues:

Solution 2 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. If you want to be clever, you don't have to do any calculation to find the eigenvalues of P.
We can double check our answer by verifying that the sum of eigenvalues equals the sum of diagonal entries: 