Question

If a, b, c are sides of a scalene triangle, then the value of $\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$ is :

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

Answer 1

The correct option is B negative

$\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$

$=a(bc-a^2)-b(b^2-ac)+c(ab-c^2)$

$=abc-a^3-b^3+abc+abc-c^3$

$=-[a^3+b^3+c^3-3abc]$

$=-(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$

$=-\frac{(a+b+c)}{2}(2a^2+2b^2+2c^2-2ab-2bc-2ca)$

$=-\frac{(a+b+c)}{2}(a^2-2ab+b^2+b^2-2bc+c^2+c^2-2ca+a^2)$

$=-\frac{a+b+c}{2}[{(a-b)}^2+{(b-c)}^2+{(c-a)}^2]$ is negative.

Hence option $\left(b\right)$ is the correct answer.