KöMaL 2013 A585

For some integer $n \ge$2 each pair ($i$,$j$), where 1$\le i$,$j \le n$, is written on a card. We play the following game. The $n^2$ cards are placed in an $n$×$n$table so that for every $i$ and $j$, the card ($i$,$j$) is in the $i$th row, at the $j$th position. It is allowed to exchange the cards ($i$,$j$) and ($k$,$l$) if they are in the same row or in the same column, and $i$=$k$ or $j$=$l$. Is it possible to reach that arrangement of the cards which contains (1,2) and (2,1) interchanged and all other cards are at their initial positions?