Question

If \(a,b,c\) are sides of a scalene triangle, then the value of \(\begin{vmatrix}a&b&c\\b&c&a\\c&a&b\end{vmatrix}\) is :

• A. non - negative
• B. negative
• C. positive
• D. non-positive

Hint:

Expand the determinant to get 3abc - (a^3 + b^3 + c^3) and use the factorisation of a^3 + b^3 + c^3 - 3abc.

Answer 1

The Correct Option is B

Expanding along the first row: \(a(cb-a^2)-b(b^2-ac)+c(ab-c^2)=3abc-(a^3+b^3+c^3)\).
Now \(a^3+b^3+c^3-3abc=\frac12(a+b+c)\left[(a-b)^2+(b-c)^2+(c-a)^2\right]\).
For sides of a scalene triangle, \(a+b+c\gt0\) and a, b, c are not all equal, so the bracket is positive. Hence \(a^3+b^3+c^3-3abc\gt0\).
So the determinant \(=-\frac12(a+b+c)\left[(a-b)^2+(b-c)^2+(c-a)^2\right]\lt0\).
Answer: (2) negative