Mathematics - Determinant Question with Solution | TestHub

$$\begin{array}{rl}& \left|\begin{array}{lll}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|=\left|\begin{array}{ccc}a+b+c& a+b+c& a+b+c\\ b& c& a\\ c& a& b\end{array}\right|\\ & =(a+b+c)\left|\begin{array}{lll}1& 1& 1\\ b& c& a\\ c& a& b\end{array}\right|\\ & =(a+b+c)\left|\begin{array}{ccc}0& 0& 1\\ b-c& c-a& a\\ c-a& a-b& b\end{array}\right|\\ & =(a+b+c)\left[ab+bc+ca-a^2-b^2-c^2\right]\\ & =-(a+b+c)\left[(a-b{)}^2+(b-c{)}^2+(c-a{)}^2\right]\end{array}$$ Since $a,b,c$ are sides of $a$ scalene triangle, therefore at least two of the $a,b,c$ will be unequal. $\begin{array}{ll}\therefore & (a-b{)}^2+(b-c{)}^2+(c-a{)}^2>0\\ & \text{ Also }a+b+c>0\\ \therefore & -(a+b+c)[(a-b{)}^2+\\ & (b-c{)}^2+(c-a{)}^2]<0\end{array}$