Is it possible to swap only two elements on a 3x3 grid by swapping rows and columns? Take a 3x3 grid with a different number in each spot, like one cell of a Sudoku puzzle:
\begin{array}{|c|c|c|}
\hline
1&2&3\\
\hline
4&5&6\\
\hline
7&8&9\\
\hline
\end{array}
Suppose you can rearrange the number by swapping two rows or two columns.
$$\left.\begin{array}{|c|c|c|}\hline 1&2&3\\ \hline 4&5&6\\ \hline7 & 8 & 9\\ \hline
\end{array} \longrightarrow \begin{array}{|c|c|c|}\hline 1&2&3\\ \hline7&8&9\\ \hline4&5&6\\ \hline
\end{array} \longrightarrow \begin{array}{|c|c|c|}\hline2&1&3\\ \hline8&7&9\\ \hline5&4&6\\ \hline
\end{array}
\right.$$
Is it possible to manipulate the grid such that an arbitrary pair of numbers is swapped (with the rest returned to their original positions)?
e.g.
$$\left.\begin{array}{|c|c|c|}\hline 1&2&6\\ \hline 4&5&3\\ \hline7&8&9\\ \hline
\end{array} \qquad\text{or}\qquad \begin{array}{|c|c|c|}\hline 8&2&3\\ \hline4&5&6\\ \hline7&1&9\\ \hline
\end{array} \qquad\text{or}\qquad \begin{array}{|c|c|c|}\hline1&2&3\\ \hline4&7&6\\ \hline5&8&9\\ \hline
\end{array}
\right.$$

Four numbers that are in the "corners" of a rectangle (like 1379 or 1278) will remain in the corners of a rectangle no matter how many or which swaps happen.  So you cannot swap just two elements while keeping the rest fixed.
