If a, b, c are sides of a scalene triangle, then the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd><mtd><mi>c</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd><mtd><mi>c</mi></mtd><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>c</mi></mtd><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd></mtr></mtable></mfenced></math> is :
Step-by-Step Solution
Key Concept: The determinant of the given circulant matrix is -(a^3 + b^3 + c^3 - 3abc) = -(a+b+c)(a^2+b^2+c^2-ab-bc-ca) = -1/2(a+b+c)((a-b)^2+(b-c)^2+(c-a)^2). Since a, b, c are sides of a triangle, a+b+c > 0 and for a scalene triangle, (a-b)^2+(b-c)^2+(c-a)^2 > 0.
Let D = |a b c; b c a; c a b|. The value of this determinant is -(a^3 + b^3 + c^3 - 3abc) = -(a+b+c)(a^2+b^2+c^2-ab-bc-ca) = -1/2(a+b+c)((a-b)^2+(b-c)^2+(c-a)^2). Since a, b, c are sides of a triangle, a+b+c > 0. Also, for a scalene triangle, a, b, and c are distinct, so (a-b)^2+(b-c)^2+(c-a)^2 > 0. Thus, the expression is negative.
Correct Answer: 2