Proving a determinant equality

Question

Prove

$$\begin{vmatrix} 2bc-a^2 & c^2 & b^2 \\ c^2 & 2ac-b^2 & b^2 \\ b^2 & a^2 & 2ab-c^2\\ \end{vmatrix} =(a^3+b^3+c^3-3abc)^2$$

My attempt:

I tried using the well-known result that

$$\begin{vmatrix} bc-a^2 & ca-b^2 & ba-c^2 \\ ac-b^2& ab-c^2 & bc-a^2 \\ ba-c^2& bc-a^2 & ca-b^2\\ \end{vmatrix} =\begin{vmatrix} a & b & c \\ b& c & a \\ c& a & b\\ \end{vmatrix}^2 =(a^3+b^3+c^3-3abc)^2$$ But I tried using many properties of determinant but I was unable to bring the l.h.s. to the form $$\begin{vmatrix} bc-a^2 & ca-b^2 & ba-c^2 \\ ac-b^2& ab-c^2 & bc-a^2 \\ ba-c^2& bc-a^2 & ca-b^2\\ \end{vmatrix}$$ Please help this one out. Thanks.

Answer 1

Hint: Observe \begin{align} \begin{bmatrix} 2bc-a^2 & c^2 & b^2\\ c^2 & 2ac-b^2 & a^2\\ b^2 & a^2 & 2ab-c^2 \end{bmatrix} = \begin{bmatrix} a & b & c\\ b & c & a\\ c & a & b \end{bmatrix} \begin{bmatrix} -a & -b & -c\\ c & a & b\\ b & c & a\\ \end{bmatrix} \end{align}

Edit: Just take the determinant.