Question

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

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

Answer 1

Answer: (B)

$\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}$
$=\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}^{2}=$ Rational & Positive