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

Hints:
(1) Determinant is a multilinear function. For example
$$\det\begin{pmatrix}a+b&c&d\\a'+b'&c'&d'\\a''+b''&c''&d''\end{pmatrix}=\det\begin{pmatrix}a&c&d\\a'&c'&d'\\a''&c''&d''\end{pmatrix}+\det\begin{pmatrix}b&c&d\\b'&c'&d'\\b''&c''&d''\end{pmatrix}$$
and likewise for each column/row.
(2) The determinant is an alternating function, i.e.: if a matrix has two equal columns/rows then its determinant is zero and if we interchange two rows/columns the determinant is multiplied by $\;-1\;$ .
