If \(a,b,c\) are positive and unequal. Define \(\triangle=\left|\begin{array}{lll}a&b&c\\ b&c&a\\ c&a&b\end{array}\right|\) which of the following is true
Answer & explanation
Correct answer: option 2
\(\triangle=\left|\begin{array}{lll}a&b&c\\ b&c&a\\ c&a&b\end{array}\right|\)
Applying $C_1→C_1+C_2+C_3$
$⇒\begin{vmatrix}a+b+c&b&c\\ a+b+c&c&a\\ a+b+c&a&b\end{vmatrix}$
Applying $R_3→R_3-R_2,\,R_2→R_2-R_1$
$(a+b+c)\begin{vmatrix}1&b&c\\0&c-b&a-c\\0&a-c&b-a\end{vmatrix}$
$=(a+b+c)(-b^2-a^2-c^2+ac+bc+ca)$
$=-\frac{1}{2}(a+b+c)((a-b)^2+(b-c)^2+(c-a)^2)<0$