Science:Math Exam Resources/Courses/MATH307/April 2005/Question 05
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Question 05 

Find for any positive integer when is the matrix 
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Hint 

Science:Math Exam Resources/Courses/MATH307/April 2005/Question 05/Hint 1 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Recall that if is diagonalisable, then . Thus, we seek the eigenvalues of to form the diagonal matrix , and the eigenvectors of to make and . To find the eigenvalues, recall that Hence we find the characteristic polynomial and set to 0: This implies that we have two eigenvalues as required: Now we must find the corresponding eigenvectors and Now we can construct and find where the columns of are the eigenvectors and the diagonal entries are the corresponding eigenvalues: It follows that 