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?

Use the rule of Sarrus to show 
 $\begin{vmatrix}0&c&b\\c&0&a\\b&a&0\end{vmatrix}^2 = (2abc)^2$ and then show that $A^2 :=\begin{pmatrix}0&c&b\\c&0&a\\b&a&0\end{pmatrix}^2=\begin{pmatrix}b^2+c^2&ab&ac\\ba&c^2+a^2&bc\\ca&cb&a^2+b^2\end{pmatrix}$ and then use $\det(A*A)=\det(A)*\det(A)$.
