Science:Math Exam Resources/Courses/MATH221/December 2008/Question 10 (b)
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Question 10 (b) 

True or false (if true, give a proof; if false, provide a counterexample): Suppose A is a 2 x 2 matrix with characteristic polynomial . Then A is diagonalizable. 
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Hint 

A matrix is diagonalizable when it has distinct eigenvalues (why is true?). 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. The statement is false. To construct a counterexample, we need to find a 2 x 2 matrix which has characteristic polynomial but has only one linearly independent eigenvector. The first matrix that comes to mind which has characteristic polynomial is the scalar multiple of the identity matrix,
However this clearly isn't a counterexample because it is a diagonal matrix. Can we modify it so that it isn't a diagonal matrix, but still has the same characteristic polynomial? A first simple modification is to investigate the diagonalizability of
Our requirement that A has only one linearly independent eigenvector means there can be only one linearly independent solution to
which is clearly the case, as is the only linearly independent eigenvector. Therefore,
is a counterexample to the statement. 