Matrices & Determinants
Circulant matrices
Grade 12
Question:
<p>If \(\alpha, \beta\) and \(\gamma\) are the roots of the equation \(x^3 + px + q = 0\), then the value of the determinant \(\begin{vmatrix} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{vmatrix}\) is</p>
<p>(a) \(a^3 + b^3 + c^2 - 3abc\)</p>
<p>(b) \(a^2b - b^2c\)</p>
<p>(c) 0</p>
<p>(d) \(a^2 + b^2 + c^2\)</p>
Step-by-Step Solution
Key Concept: Recognize the circulant matrix structure and use Vieta's formulas for the cubic equation.
<p>The determinant is a circulant matrix form. Using properties of circulant matrices and the fact that \(\alpha + \beta + \gamma = 0\) (coefficient of \(x^2\) in the cubic is 0), the determinant evaluates to 0.</p>
Correct Answer: C