Definition: The permutation matrix $P_{ij}$ is the identity matrix with rows $i$ and $j$ reversed. When left-multiplied with another matrix, it exchanges rows $i$ and $j$.
Am I right in thinking that $P_{12} = P_{21}$ because, in both permutation matrices, rows 1 and 2 are reversed?
In other words, does:
$$P_{12} \stackrel{?}{=} P_{21} \stackrel{?}= \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} $$
Answer: Yes.
Yes, Jason, your intuition is correct:
$$P_{12} = P_{21}= \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} $$
just as in cyclic notation, we have the transposition $(1 \;\;2) = (2\;\; 1)$