Question

If $a,b,c$ are positive and unequal, show that value of the determinant $\Delta =\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$ is negative.

Answer 1

$R_1\to R_1+R_2+R_3$ is $△=\left|\begin{array}{ccc}a+b+c& a+b+c& a+b+c\\ b& c& a\\ c& a& b\end{array}\right|$

$\Rightarrow △=(a+b+c)$

$=\left|\begin{array}{ccc}1& 1& 1\\ b& c& a\\ c& a& b\end{array}\right|$

$C_1\to C_1-C_1$ $C_2\to C_2-C_3$

$\Rightarrow △=\left|\begin{array}{ccc}0& 0& 1\\ b-c& c-a& a\\ c-a& a-b& b\end{array}\right|(a+b+c)$

$△=(a+b+c)\left(\right(b-c\left)\right(a-b)-(c-a\left)\right(c-a\left)\right)$

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

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

$=-(a^3+b^3+c^3)$

$=negative$

$(\because a^3+b^3+c^3>0)$

Hence proved.