Science:Math Exam Resources/Courses/MATH307/April 2010/Question 04 (b)
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Question 04 (b) 

Consider the following recursion relation: where , and How must the parameters a and b be related if it is given that has as its eigenvalue? What is the other eigenvalue of (in terms of a and b )? 
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 2010/Question 04 (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. Knowing has an eigenvalue of 1 we need to solve the equation: where =
making it into a system of linear equations we get: 1) = 2) = using 2) we can substitute into 1) so that we are only working with one variable, = ( )
now that we see how and have to relate, let’s find the other eigenvalue of solve for either or and input back into and then do our standard I’m going to continue after having solved for so our new =
we end up obtaining the equation: ( ) using the quadratic formula with a = , b = and c = , our result is:
, we knew about the eigenvalue of already, and so the other eigenvalue of is or, equivalently, 