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

Every invertible matrix can be diagonalized. 
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Hint 

Science:Math Exam Resources/Courses/MATH221/April 2009/Question 12 (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. This is false. To construct a counterexample, we need to think of an invertible matrix which cannot be diagonalized, i.e. an invertible matrix which has only one linearly independent eigenvector. The first invertible matrix which comes to mind is the identity matrix,
however, this is not a counterexample as it is diagonal. However, consider the simple modification
A is invertible as its determinant is 1, however, A is not diagonalizable as the only linearly independent solution to the eigenvector equation
is , and so A has only one linearly independent eigenvector corresponding to its eigenvalue 1. Therefore, we conclude is a counterexample to the statement. 