Prove the following determinant.

Question

Prove that : $$\left| \begin {array} c (2bc-a^2) & c^2 & b^2 \\ c^2 & (2ac-b^2) & a^2 \\ b^2 & a^2 & (2ab-c^2) \end{array} \right| =D^2.$$

Where $$D= \left| \begin {array} c a& b &c \\ b & c & a \\ c& a & b \end {array} \right|.$$

It is my problem .I simply expand it but it made complex so I think we need to use several operations.

I edited the question to properly format the first determinant; please tell me if I have made any mistakes when converting into TeX. — FraGrechi, Mar 08 '17 at 15:57
Is $D$ as what it is? Is the last element in the second row $b$? — , Mar 08 '17 at 16:07

score 2 · Accepted Answer · answered Mar 08 '17 at 16:47

Write $D^2$ as $ \begin{vmatrix} b & c & a\\ c & a & b\\ a & b & c \end{vmatrix} \times \begin{vmatrix} c & a & b\\ b & c & a\\ -a & -b & -c \end{vmatrix} $

Proving that $D$ is equivalent to each of the above two determinants is a trivial exercise

Prove the following determinant.

1 Answers1