Proving determinants' property $$\begin{vmatrix}
z & x & c  \\
a+b & a+b & a+b \\
q & w & e  \\
\end{vmatrix} =
\begin{vmatrix}
z & x & c  \\
a & a & a \\
q & w & e  \\
\end{vmatrix}+\begin{vmatrix}
z & x & c  \\
b & b & b \\
q & w & e  \\
\end{vmatrix}$$
I think this is called the multilinearity of determinants. How do I prove it? I tried to expand using the definition of determinants yet how is it that when the terms are the same, the addition does not produce 2 times of that unchanged terms?

Expand all the three determinants along the middle row. Let $D_1=xe-cw$, $D_2=ze-cq$ and $D_3=zw-xq$:
$$(a+b)(-D_1+D_2-D_3)=a(-D_1+D_2-D_3)+b(-D_1+D_2-D_3)$$
which is obviously true.
The same identity holds for $n\times n$ matrices, just call $D_1,\dots,D_n$ the relevant determinants and you have the same proof, so nothing particular about $n=3$.
