Science:Math Exam Resources/Courses/MATH152/April 2010/Question A 05

MATH152 April 2010

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Question A 05
If 2 and 3 are both eigenvalues of a $2\times {2}$ matrix A then A must be invertible. Justify this statement briefly.

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
How many eigenvalues does an $n\times {n}$ matrix have. How do the eigenvalues relate to the invertibility of a matrix?

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Recall that $n\times {n}$ matrices have n eigenvalues. Here since n=2 we have 2 eigenvalues which are given as 2 and 3. Also recall that if A is not invertible then there exists at least one non-zero vector x such that Ax=0. Therefore, we see that if A is not invertible then there is at least one zero eigenvalue with eigenvector x. Conversely, if 0 is an eigenvalue then there exists a non-zero vector x such that Ax=0 which means A is not invertible. This concludes that A being non-invertible implies a zero eigenvalue and also that a zero eigenvalue implies that A is non-invertible. We just wrote that the only eigenvalues are 2 and 3 so therefore, 0 is not an eigenvalues which means A must be invertible.

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