Science:Math Exam Resources/Courses/MATH152/April 2011/Question A 26

MATH152 April 2011

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Question A 26
For which values of the parameters $\displaystyle a,b,c$ is the following matrix invertible? $B={\begin{bmatrix}a&a&a\\a&a&b\\a&b&c\end{bmatrix}}$

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?

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Hint
What is an easy to compute value that determines when a matrix is invertible?

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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. We compute the determinant of our matrix $\displaystyle {\begin{aligned}\det(B)&=(a^{2}c+a^{2}b+a^{2}b-(a^{3}+a^{2}c+ab^{2}))\\&=2a^{2}b-a^{3}-ab^{2}\end{aligned}}$ In order for the matrix to be invertible, we need the determinant to be nonzero. For that, we need to factor our determinant $\displaystyle {\begin{aligned}\det(B)&=2a^{2}b-a^{3}-ab^{2}\\&=-a(a^{2}-2ab+b^{2})\\&=-a(a-b)^{2}\end{aligned}}$ So we obtain that $\displaystyle \det(B)=-a(a-b)^{2}$ and so, for the matrix to be invertible, that is, for the matrix not to have a determinant of zero, we require that $a\neq 0$ and $a\neq b$.