Question

If a, b, c are all positive and not all equal 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. cannot be determined

Answer 1

The correct option is B < 0

Let $|A|=\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$

$C_1\to C_1+C_2+C_3$
$|A|=\left|\begin{array}{ccc}a+b+c& b& c\\ a+b+c& c& a\\ a+b+c& a& b\end{array}\right|$

$=(a+b+c)\left|\begin{array}{ccc}1& b& c\\ 1& c& a\\ 1& a& b\end{array}\right|$
$R_1\to R_1-R_2,R_2\to R_2-R_3$
$=(a+b+c)\left|\begin{array}{ccc}0& b-a& c-a\\ 0& c-a& a-b\\ 1& a& b\end{array}\right|$

$=(a+b+c)\left[\right(b-a\left)\right(a-b)-(c-a{)}^2]$
$\therefore |A|=-(a+b+c)\left[\right(a-b{)}^2+(c-a{)}^2]$

Now, $(a-b{)}^2,(c-a{)}^2>0$ and $a+b+c>0$ as $a,b,c>0$

$\therefore |A|<0$

Hence, $\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|<0$