Matrices & Determinants
Circulant Matrices
Grade 12
Question:
<p>If \(a\), \(b\) and \(c\) are the roots of the equation \(x^3 + 2x^2 + 1 = 0\), find <span class="math">\begin{vmatrix} a & b & x \\ b & c & a \\ c & a & b \end{vmatrix}</span>.</p>
<p>(a) 8</p>
<p>(b) -8</p>
<p>(c) 0</p>
<p>(d) 2</p>
Step-by-Step Solution
Key Concept: Recognize circulant matrix structure and apply properties of roots via Vieta's formulas.
<p><strong>Solution:</strong> This is a circulant determinant. Use properties of circulant matrices and Vieta's formulas for the roots of $x^3 + 2x^2 + 1 = 0$. The determinant evaluates to 8.</p>
Correct Answer: A