Question

If $a,b,c$ are all positive and unequal numbers, then using the properties of determinants, prove that the value of the determinant $\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$ is negative.

Answer 1

$a,b,c$ are positive and unequal

$\left[\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right]$

$=a(bc-a^2)+b(ac-b^2)+c(ab-c^2)=abc-a^3+abc-b^3+abc-c^3=-(a^3+b^3+c^3-3abc)=-(a+b+c)(a^2+b^2+c^2-ab-bc-ca)=-(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$

As, $a,b$ and $c$ are positive and unequal

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

Both terms are positive so solution is negative as $(-1)$ sign is present