example of permutation matrix

Consider the matrix

that corresponds to permuting the columns of the identity matrix under the permutation $(1243)$ (i.e. (http://planetmath.org/Ie), the first column of the identity matrix is the second column of $P$, the second column of the identity matrix is the fourth column of $P$, etc.). Then $P$ is a permutation matrix.

We will consider what happens when we multiply a $4\times 4$ matrix by $P$. For example, let $A$ be the matrix

Then

We notice that $PA$ has the same rows as $A$. Moreover, the rows of $A$ are the rows of $PA$ permuted under $(1243)$: The first row of $PA$ is the second row of $A$, the second row of $PA$ is the fourth row of $A$, etc.