Question

Show that the determinant
$\left|\begin{array}{ccc}{a}^2+b^2+c^2& bc+ca+ab& bc+ca+ab\\ bc+ca+ab& a^2+b^2+c^2& bc+ca+ab\\ bc+ca+ab& bc+ca+ab& a^2+b^2+c^2\end{array}\right|$
is always non-negative.

Answer 1

$\left|\begin{array}{ccc}{a}^2+b^2+c^2& bc+ca+ab& bc+ca+ab\\ bc+ca+ab& a^2+b^2+c^2& bc+ca+ab\\ bc+ca+ab& bc+ca+ab& a^2+b^2+c^2\end{array}\right|=\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|={\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|}^2={(a+b+c)}^2{(a^2+b^2+c^2-ab-bc-ca)}^2>0$

for all $a,b,c,a\ne b\ne c$

$\therefore$ it always non-negative.(positive).