I'm positive there's a sufficient answer for this here, but I can't seem to find it. I'd appreciate being redirected to the original question if anyone knows of one.

Take the augmented matrix below as an example, $$ \pmatrix{a_1 \ a_2 \ a_3 \\ b_1 \ b_2 \ b_3 \\ c_1\ c_2\ c_3}$$ Why is it that I am allowed to take the first component $a_1$, cover up every other element corresponding to its row and column, and multiply it to the remaining matrix $\pmatrix{b_1 \ b_3 \\ c_2 \ c_3}$, and then repeat this toward the last element on the first row? And why do the signs alternate? Like so: $$a_1\pmatrix{b_2 \ b_3 \\ c_2 \ c_3}-a_2\pmatrix{b_1 \ b_3 \\ c_1 \ c_3}+a_3\pmatrix{b_1 \ b_2 \\ c_1 \ c_2}=$$