Matrices & Determinants
3×3 Determinants
Grade 12
Question:
<p>If <i>a</i> = α<sup>2</sup> + β<sup>2</sup> + γ<sup>2</sup>, <i>b</i> = αβ + βγ + γα, the value of <span style='display:inline-block'>\[\begin{vmatrix} b & a & b \\ b & a & b \\ b & b & a \end{vmatrix}\]</span> is</p>
<p>(a) 14</p>
<p>(b) 49</p>
<p>(c) 98</p>
<p>(d) 196</p>
Step-by-Step Solution
Key Concept: Express <i>a</i> and <i>b</i> in terms of Vieta's formulas, then recognize the determinant pattern to compute its value.
<p><strong>Solution:</strong> Using α + β + γ = −2 and αβ + βγ + γα = −1, compute <i>a</i> = (α + β + γ)<sup>2</sup> − 2(αβ + βγ + γα) = 4 − 2(−1) = 6 and <i>b</i> = −1. The determinant becomes a special form that evaluates to 196.</p>
Correct Answer: d