Determinant without computing How can I show without computing the determinant that the equation is true?
$$\det \begin{pmatrix}
b1 + c1 & c1 + a1 & a1 + b1\\
b2 + c2 & c2 + a2 & a2 + b2\\
b3 + c3 & c3 + a3 & a3 + b3
\end{pmatrix} = 2 \det \begin{pmatrix}
a1 & b1 & c1\\
a2 & b2 & c2\\
a3 & b3 & c3
\end{pmatrix}$$

Try expanding by multilinearity. The determinant is linear in each column (or in each row, if you prefer).
