Question

If $a,b,c$ are distinct and rational numbers then $\left|\begin{array}{ccc}\left(a^2+b^2+c^2\right)& ab+bc+ca& ab+bc+ca\\ ab+bc+ca& \left(a^2+b^2+c^2\right)& \left(ab+bc+ca\right)\\ ab+bc+ca& \left(ab+bc+ca\right)& \left(a^2+b^2+c^2\right)\end{array}\right|$ is always

• A. zero
• B. Rational & Positive
• C. Rational & Negative
• D. Irrational and Positive

Answer 1

Answer: (B)

$\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$
$={\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|}^2=$ Rational & Positive