Matrices & Determinants
Special Determinants
Grade 12
Question:
<p>The value of <span style='display:inline-block'>\[\begin{vmatrix} \alpha & \beta & \gamma \\ \gamma & \alpha & \beta \\ \beta & \gamma & \alpha \end{vmatrix}\]</span> where α, β, γ are the roots of <i>x</i><sup>3</sup> + 2<i>x</i><sup>2</sup> − <i>x</i> − 3 = 0 is equal to</p>
<p>(a) 14</p>
<p>(b) −2</p>
<p>(c) 10</p>
<p>(d) 14</p>
Step-by-Step Solution
Key Concept: Recognize circulant matrix structure and apply Vieta's formulas for roots of the cubic to compute the determinant.
<p><strong>Solution:</strong> For a circulant determinant with roots α, β, γ of the cubic equation, use Vieta's formulas: α + β + γ = −2, αβ + βγ + γα = −1, αβγ = 3. The determinant evaluates to 14.</p>
Correct Answer: d