Science:Math Exam Resources/Courses/MATH221/April 2009/Question 12 (i)
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Question 12 (i) 

There is no matrix A with eigenvectors , and with corresponding eigenvalues 1, 1, 4. 
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Hint 

Science:Math Exam Resources/Courses/MATH221/April 2009/Question 12 (i)/Hint 1 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. The answer is false. Let and denote the given vectors. A matrix A has eigenvectors with corresponding respective eigenvalues 1, 1, 4 if and only if , where and Thus the matrix A exists if and only if P is invertible (that is, if and only if are linearly independent). We can calculate . Therefore, P is invertible and the desired matrix A will be given by . Remark: By computing the inverse (not necessary to answer the question), we can determine the explicit form of A as
