<p>If <i>a</i>, <i>b</i>, <i>c</i> are sides of a scalene triangle, then the value of determinant \(\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}\) is always:</p>
Step-by-Step Solution
Key Concept: The determinant of a cyclic matrix can be factored using the identity that relates it to (a+b+c) and a quadratic form. For a scalene triangle, we must use the triangle inequality to determine the sign of this determinant.
<p><strong>Step 1:</strong> Calculate the determinant using the rule for 3×3 matrices or recognize this as a cyclic matrix.</p><p>Let Δ = |a b c; b c a; c a b|</p><p>Expanding: Δ = a(ca - ab) - b(ba - ac) + c(ba - c²)</p><p>Δ = a²c - a²b - b²a + abc + abc - c³</p><p>Δ = a²c - a²b - ab² + 2abc - c³</p><p><strong>Step 2:</strong> Use the known factorization for cyclic determinants:</p><p>Δ = (a + b + c)(ab + bc + ca - a² - b² - c²)</p><p>Which can be rewritten as:</p><p>Δ = -(a + b + c)[(a-b)² + (b-c)² + (c-a)²]/2</p><p><strong>Step 3:</strong> Analyze the sign. Since a, b, c are sides of a triangle: a + b + c > 0 (always positive).</p><p><strong>Step 4:</strong> For a scalene triangle, a ≠ b, b ≠ c, c ≠ a, so (a-b)² + (b-c)² + (c-a)² > 0 (strictly positive).</p><p><strong>Step 5:</strong> Therefore: Δ = -(positive)(positive) < 0</p><p>The determinant is always NEGATIVE for a scalene triangle.</p><p><strong>∴ Answer:</strong> D</p>
Correct Answer: D