Science:Math Exam Resources/Courses/MATH307/April 2009/Question 08 (b)
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Question 08 (b) 

A player begins a game of chance by placing a marker in box 2, marked start. A die is rolled, and the marker is moved one square to the left if a 1 or 2 is rolled and one square to the right if a 3, 4, 5, or 6 is rolled. This process continues until the marker lands in square 1, in which case the player wins the game, or in square 4, in which case the player loses the game. (b) How would you check in MATLAB/Octave whether the above stochastic matrix has only one eigenvalue equal to 1 or multiple eigenvalues equal to 1? Write down the commands you would use and explain how the output would indicate which of the possibilities is true. 
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 2009/Question 08 (b)/Hint 1 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Input the following: P = [1,1/3,0,0;0,0,1/3,0;0,2/3,0,0;0,0,2/3,1]; eigs(P) The eigs command gives us a column vector listing the eigenvalues of P, including any repetitions. As such, if the value 1 appears more than once in the output vector, then P has multiple eigenvalues equal to 1. 