Why is the permutation matrix called so? Any combinatorial meaning? My question is very simple but I cannot really have it answer. Why the permutation matrix is called permutation matrix?? Is there any combinatorial meaning with the permutation matrix? (As I know, a permutation matrix have one and all zero for all entries of every rows and columns)
 A: Try writing down such a matrix in a small dimension, and see how it acts on the standard basis.  Say
$$A = \left(\begin{array}{rrr}
0 & 1 & 0\\
0  & 0 & 1\\
1 & 0 & 0
\end{array}\right).$$
Then
$$\left(\begin{array}{rrr}
0 & 1 & 0\\
0  & 0 & 1\\
1 & 0 & 0
\end{array}\right)
\left(\begin{array}{c}
1 \\
0\\
0 
\end{array}\right)
=\left(\begin{array}{c}
0 \\
0\\
1 
\end{array}\right),
$$
$$\left(\begin{array}{rrr}
0 & 1 & 0\\
0  & 0 & 1\\
1 & 0 & 0
\end{array}\right)
\left(\begin{array}{c}
0 \\
1\\
0 
\end{array}\right)
=\left(\begin{array}{c}
1 \\
0\\
0 
\end{array}\right),
$$
$$\left(\begin{array}{rrr}
0 & 1 & 0\\
0  & 0 & 1\\
1 & 0 & 0
\end{array}\right)
\left(\begin{array}{c}
0 \\
0\\
1 
\end{array}\right)
=\left(\begin{array}{c}
0 \\
1\\
0 
\end{array}\right),
$$
So you see, they are called "permutation matrices" because they permute the standard basis.  In fact, this example I wrote down is the standard representation of $S_3$ (the permutation group on $3$ letters) in $GL_3$ (the group of invertible $3\times3$ matrices).
Once you know they permute the standard basis, it follows that multiplying a matrix on the left by a permutation matrix will permute the rows, and multiplying on the right by a permutation matrix will permute the columns.
A: If you multiply any matrix at the right/left by a permutation matrix, you simply permute the rows and respectively columns of that matrix by the permutation $\sigma$, respectively by $\sigma^{-1}$.
