Question

If $a,b,c$ be three distinct positive 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. negative
• B. positive
• C. zero
• D. not possible to determine

Answer 1

The correct option is A negative

$\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|=-(a^3+b^3+c^3-3abc)$
$=-(a+b+c)(a^2+b^2+c^2-ab-bc-ca)=-\frac{1}{2}(a+b+c)\left[\right(a-b{)}^2+(b-c{)}^2+(c-a{)}^2]$
As all the numbers are positive, so clearly the determinant is negative.