Finding the determinant of a matrix given by three parameters. 
Show that for $a,b,c\in\mathbb R$ $$\begin{vmatrix}b^2+c^2&ab&ac\\ba&c^2+a^2&bc\\ca&cb&a^2+b^2\end{vmatrix} = \begin{vmatrix}0&c&b\\c&0&a\\b&a&0\end{vmatrix}^2 = 4a^2b^2c^2. $$

There must be some trick, like using elementary row operations, to get the determinant into that form, but I am not seeing it. And directly computing the determinant by the cofactor expansion looks very nasty. So is there a simpler way to compute this determinant?

A determinant of the left matrix is equal to
$$\prod_{cyc}(a^2+b^2)+2a^2b^2c^2-\sum_{cyc}(a^2+b^2)a^2b^2=$$
$$=\sum_{cyc}\left(a^4b^2+a^4c^2+\frac{2}{3}a^2b^2c^2\right)+2a^2b^2c^2-\sum_{cyc}(a^4b^2+a^4c^2)=4a^2b^2c^2.$$
A determinant of the right matrix is equal to
$$0+2abc-0=2abc$$ and we are done!
