show that the determinant
`|{:(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)):}|`
is always non- negative.

Explanation

`|{:(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)):}|`
`=|underset(c" "a" "b)underset(b" "c" "a)(a" "b" "c)||underset(c" "a" "b)underset(b" "c" "a)(a" "b" "c)|`
Which is always non-negative.