Science:Math Exam Resources/Courses/MATH152/April 2010/Question A 06

MATH152 April 2010

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Question A 06
For what values of the parameter a does the matrix ${\begin{bmatrix}1&0&a\\-1&1&1\\2&1&0\end{bmatrix}}$ not have an inverse?

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 do we relate information about inverses to determinants?

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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. A requirement for a matrix to be invertible is that its determinant is non-zero. Conversely, non-invertible matrices have zero determinants. Therefore, if we compute the determinant of the matrix and force it to be zero, that will give a requirement on a. We can compute the determinant of the matrix by cofactor expansion (we will choose the first row), ${\begin{vmatrix}1&0&a\\-1&1&1\\2&1&0\end{vmatrix}}=1{\begin{vmatrix}1&1\\1&0\end{vmatrix}}+a{\begin{vmatrix}-1&1\\2&1\end{vmatrix}}=-3a-1.$ We want this determinant to vanish and so we require -3a-1=0 or a=-1/3. Therefore, if a=-1/3 then the matrix will have a zero determinant and thus not be invertible.