Let a, b, c be positive and not all equal, the value of the determinant $\begin{vmatrix}a &b &c\\b &c&a\\c&a &b\end{vmatrix}$, is

- ive

Let $Δ=\begin{vmatrix}a &b &c\\b &c&a\\c&a &b\end{vmatrix}$. Then,

$Δ=\begin{vmatrix}a+b+c &b &c\\b+c+a &c&a\\c+a+b&a &b\end{vmatrix}$ [Applying $C_1→C_1+C_2 +C_3$]

$⇒Δ=(a+b+c)\begin{vmatrix}1 &b &c\\1 &c&a\\1&a &b\end{vmatrix}$  [Taking (a+b+c) common $C_1$]

$⇒Δ=(a+b+c)\begin{vmatrix}1 &b &c\\0 &c-b&a-c\\0&a-b &b-c\end{vmatrix}$ [Applying $R_2→ R_2-R_1,R_3→ R_3-R_1$]

$⇒Δ=(a+b+c)1\begin{vmatrix}c-b&a-c\\a-b&b-c\end{vmatrix}$  [Expanding along $C_1$]

$⇒Δ=(a+b+c)(-(b-c)^2- (a-c) (a-b))$

$⇒Δ=-(a+b+c)[(b-c)2 + (a-c) (a - b)]$

$⇒Δ=-\frac{1}{2}(a+b+c)\{(a - b)^2 + (b-c)^2+(c-a)^2\} <0$  [∵$a≠b≠c$ and $a, b, c>0$]