Question

Let $a,b,c$ be positive and not all equal. Show that the value of the determinant $\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$ is negative.

Answer 1

Expanding along $R_1$

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

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

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

$=-\left[\right(a+b+c\left)\right(a^2+b^2+c^2-ab-bc-ca\left)\right]$

$=-\frac{1}{2}\left[\right(a+b+c\left)\right(2a^2+2b^2+2c^2-2ab-2bc-2ca\left)\right]$

$=-\frac{1}{2}\left[\right(a+b+c\left)\right((a-b{)}^2+(b-c{)}^2+(c-a{)}^2)]$

$=-ve$ (therefore, $a,b,c$ are positive and distinct)