A common notation for a permutation $\pi$ of the integers $\{1, \ldots, m\}$ is as a $2 \times m$ $\operatorname{matrix}\left(\begin{array}{ccccc}1 & 2 & 3 & \ldots & m \\ \pi(1) & \pi(2) & \pi(3) & \ldots & \pi(m)\end{array}\right)$, indicating that $\pi$ takes $i$ to $\pi(i)$. (a) Show that such a permutation corresponds to the permutation matrix with 1's in positions $(\pi(j), j)$ for $j=1, \ldots, m$. (b) Write down the permutation matrices corresponding to the following permutations:
(i) $\left(\begin{array}{lll}1 & 2 & 3 \\ 2 & 1 & 3\end{array}\right)$,
(ii) $\left(\begin{array}{llll}1 & 2 & 3 & 4 \\ 4 & 2 & 3 & 1\end{array}\right)$,
(iii) $\left(\begin{array}{llll}1 & 2 & 3 & 4 \\ 1 & 4 & 2 & 3\end{array}\right)$, (iv) $\left(\begin{array}{lllll}1 & 2 & 3 & 4 & 5 \\ 5 & 4 & 3 & 2 & 1\end{array}\right)$. Which are elementary matrices? (c) Write down, using the preceding notation, the permutations corresponding to the following permutation matrices:
(i) $\left(\begin{array}{lll}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{array}\right)$,
(ii) $\left(\begin{array}{llll}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\end{array}\right)$,
(iii) $\left(\begin{array}{llll}0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0\end{array}\right)$,
(iv) $\left(\begin{array}{lllll}0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0\end{array}\right)$.