Science:Math Exam Resources/Courses/MATH221/December 2008/Question 05 (b)
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Question 05 (b) 

Let . Find the matrix . (Simplify your answer as much as possible.) 
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 

Make use of the result of part (a) to diagonalize . Powers of diagonal matrices are easy to compute. 
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. From part (a), we know that the eigenvalues for are and , with corresponding eigenvectors and , respectively. Let denote the matrix of eigenvalues, and the matrix whose columns are the corresponding eigenvectors. Then is diagonalized via , or, . This decomposition allows for easy computations of powers of , as for any integer , we have , and . Therefore, we compute , and multiply the three matrices to obtain
Then for , we have 