Question 48
$n^2$ real numbers are written in a square $n \times n$ table so that the sum of the numbers in each row and column equals zero. A move is to add a row to one column and subtract it from another (so if the entries are $a_{ij}$ and we select row $i$, column $h$ and column $k$, then column $h$ becomes $a_{1h} + a_{i1}$, $a_{2h} + a_{i2}$, ..., $a_{nh} + a_{in}$, column $k$ becomes $a_{1k} - a_{i1}$, $a_{2k} - a_{i2}$, ..., $a_{nk} - a_{in}$, and the other entries are unchanged). Show that we can make all the entries zero by a series of moves.