Leibniz formula for determinants I don't fully understand the formula for Leibniz formula of determinant. It is in my book as:\
$\ \Sigma sgn(\pi )A_{1\pi (1)}...A_{n\pi (n)}$ 
I kind of understand that it has to do with eradicating repeated rows and the amount of rows that have been interchanged but that is about the extent of it. An example of a 3x3 matrix would be great, thanks.
 A: Let
$$A=\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}.$$
Using first row expansion, we get
$$\det(A)=a_{11}\det\begin{pmatrix}a_{22}&a_{23}\\a_{32}&a_{33}\end{pmatrix}-a_{12}\det\begin{pmatrix}a_{21}&a_{23}\\a_{31}&a_{33}\end{pmatrix}+a_{13}\det\begin{pmatrix}a_{21}&a_{22}\\a_{31}&a_{32}\end{pmatrix}\\
=a_{11}a_{22}a_{33}-a_{11}a_{23}a_{32}-a_{12}a_{21}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{13}a_{22}a_{31}.$$
From the above formula, each term is in the form of $a_{1j_1}a_{2j_2}a_{3j_3}$ where the second indices $j_1, j_2, j_3$ is a permutation of $1,2,3$. For example, the first term has $1,2,3$ as the second indices; the second term has $1,3,2$; the third term has $2,1,3$, etc. 
The sign of each term is the sign of the permutation. For example, the sign of $1,2,3$ is clearly $1$. For the second term, $1,3,2$ is obtained from $1,2,3$ by one transposition $2,3\rightarrow 3,2$, so the sign is $-1$, etc. 
Generalizing this to $n\times n$ matrices gives the formula $\sum_{\pi}\text{sign}(\pi)a_{1\pi(1)}\cdots a_{n\pi(n)}$.
