Question

If a$\ne$ b $\ne$ c are all positive, then the value of the determinants $\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$ is.

• A. Non-negative
• B. Non-positive
• C. Negative
• D. Positive

Answer 1

The correct option is C Negative

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

$\Rightarrow a(cb-a^2)-b(b^2-ac)+c(ab-c^2)$

$\Rightarrow 3abc-a^3-b^3-c^3$

$\because -(a^3+b^3+c^3-3acb)$

$=-\left((a+b+c)(a^2+b^2+c^2-ab-bc-ac)\right)$

$=-\frac{1}{2}\left((a+b+c)[{(a-b)}^2+{(b-c)}^2+{(c-a)}^2]\right)$ }-positive

Determinant value is negative

Option C