If a, b, c are sides of a scalene triangle, then the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|
Step-by-Step Solution
Key Concept: The determinant of the circulant matrix |a b c; b c a; c a b| is -(a+b+c)(a^2+b^2+c^2-ab-bc-ca). Since a, b, c are sides of a scalene triangle, a+b+c > 0 and a^2+b^2+c^2-ab-bc-ca = 1/2((a-b)^2+(b-c)^2+(c-a)^2) > 0 because the triangle is scalene (not all sides are equal). Thus, the expression is negative.
The determinant is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>)</mo><mo>(</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><msup><mi>c</mi><mn>2</mn></msup><mo>-</mo><mi>a</mi><mi>b</mi><mo>-</mo><mi>b</mi><mi>c</mi><mo>-</mo><mi>c</mi><mi>a</mi><mo>)</mo></math>. Since a, b, c are sides of a triangle, a+b+c > 0. For a scalene triangle, a, b, c are not all equal, so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><msup><mi>c</mi><mn>2</mn></msup><mo>-</mo><mi>a</mi><mi>b</mi><mo>-</mo><mi>b</mi><mi>c</mi><mo>-</mo><mi>c</mi><mi>a</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>[</mo><mo>(</mo><mi>a</mi><mo>-</mo><mi>b</mi><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><mo>(</mo><mi>b</mi><mo>-</mo><mi>c</mi><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><mo>(</mo><mi>c</mi><mo>-</mo><mi>a</mi><msup><mo>)</mo><mn>2</mn></msup><mo>]</mo><mo>></mo><mn>0</mn></math>. Therefore, the value of the determinant is negative.
Correct Answer: (B)