Prove the determinant of A equals 2 times the determinant of B

Question

Question: Prove that

$$ det \begin{bmatrix} a + b & p + q & u + v \\ b + c & q + r & v + w \\ c + a & r + p & w + u \\ \end{bmatrix} =2det \begin{bmatrix} a & p & u \\ b & q & v \\ c & r & w \\ \end{bmatrix} $$

I believe it's related to row operations with determinants but I have no idea where to start. Any ideas?

Answer 1

Hint: try the operation "replace the first line $L_1$ by $L_1-L_2+L_3$" and see what it gives you.