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. Thus the product is negative.
The determinant is given by -(a+b+c)(a^2+b^2+c^2-ab-bc-ca). For a scalene triangle, a, b, c are distinct positive real numbers such that the sum of any two is greater than the third. The expression a^2+b^2+c^2-ab-bc-ca = 1/2((a-b)^2+(b-c)^2+(c-a)^2) is strictly positive because a, b, c are distinct. Thus, the determinant is negative.
Correct Answer: B