Question

If $a,b,c$ are distinct positive real numbers, then the value of the determinant $\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$ is

• A. < 0
• B. > 0
• C. 0
• D. $\ge 0$

Answer 1

Answer: (A)

Let $A=\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$
$[$ apply $C_1\to C_1+C_2+C_3]$
$=\left|\begin{array}{ccc}a+b+c& b& c\\ a+b+c& c& a\\ a+b+c& a& b\end{array}\right|$
$=(a+b+C)\left|\begin{array}{ccc}1& b& c\\ 1& c& a\\ 1& a& b\end{array}\right|$
$[$ apply $R_2\to R_2-R_1$ and $R_3\to R_3-R_1]$
$=(a+b+c)\left|\begin{array}{ccc}1& b& c\\ 0& c-b& a-c\\ 0& a-b& b-c\end{array}\right|$
$=(a+b+c)[(c-b)(b-c)-(a-c)(a-b)]$
$=(a+b+c)[bc-b^2-c^2+bc-a^2+ab+ac-bc]$
$=-(a+b+c)\left(a^2+b^2+c^2-ab-bc-ca\right)$
$=-\frac{1}{2}(a+b+c)\left[(a-b{)}^2+(b-c{)}^2+(c-a{)}^2\right]$
Since, $a,b,c$ are distinct positive numbers.
$\therefore$ The value of determinant $A$ is less than $0$ .