Science:Math Exam Resources/Courses/MATH307/December 2010/Question 05 (c)
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Question 05 (c) 

Define a sequence x_{0}, x_{1}, ... by the initial conditions x_{0} = a, x_{1} = b and x_{2} = c together with the recursion relation for n = 0, 1, 2, ... (c) Explain how you could ensure that the a, b and c you find in part (b) are real numbers. 
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 

For a real matrix, if is an eigenvector to a complex eigenvalue , then is an eigenvector to a complex eigenvalue 
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. For real matrices, just as complex eigenvalues, also complex eigenvectors come in conjugate pairs. Hence Hence v_{2} + v_{3} is a real vector and therefore we ensure that the solution vector [c, b, a]^{T} in part (b) is real by choosing real coefficients s and t and setting s = t: 