Matrices & Determinants
Determinant as Polynomial
Grade 12
Question:
<p>If one of the roots of the equation <span class="math">\begin{vmatrix} 7 & 6 & x^2-13 \\ 2 & x^2-13 & 2 \\ x^2-13 & 3 & 7 \end{vmatrix} = 0</span> is \(x = 2\), then the sum of all other five roots is:</p>
<p>(a) -2</p>
<p>(b) 0</p>
<p>(c) 2\(\sqrt{5}\)</p>
<p>(d) 1\(\sqrt{5}\)</p>
Step-by-Step Solution
Key Concept: Expand the determinant to get a polynomial equation; use the given root and Vieta's formulas to find the sum of remaining roots.
<p><strong>Solution:</strong> Expand the determinant to obtain a polynomial equation in $x$. Since the determinant involves $(x^2 - 13)$ terms, this is a 6th-degree polynomial in $x$ (or equivalently, a cubic in $x^2$). Given that $x = 2$ is a root (so $x^2 = 4$), find the other roots using polynomial properties and sum them appropriately. The sum of the other five roots is -2.</p>
Correct Answer: A