Question

If a,b,c be positive and not all equal, then the value of the determinant $\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$ is

• A. –ve
• B. + ve
• C. Depends on a, b, c
• D. None of these

Answer 1

The correct option is A –ve

$\Delta =-(a^3+b^3+c^3-3abc)=-(a+b+c)(a^2+b^2+c^2-ab-bc-ca)=-\frac{1}{2}(a+b+c)\left[\right(a-b{)}^2+(b-c{)}^2+(c-a{)}^2],$.
which is clearly negative because of the given conditions.