RU-ALLRUSSIAN 1988 Problem 16

$\mathrm{n}^{2}$ real numbers are written in a square $\mathrm{n}\times \mathrm{n}$ 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 $\mathrm{a_{ij}}$ and we select row i, column h and column k, then column h becomes $\mathrm{a_{1h} + a_{i1}}$ , $\mathrm{a_{2h} + a_{i2}}$ , ..., $\mathrm{a_{nh} + a_{in}}$ , column k becomes $\mathrm{a_{1k} - a_{i1}}$ , $\mathrm{a_{2k} - a_{i2}}$ , ..., $\mathrm{a_{nk} - a_{in}}$ , and the other entries are unchanged). Show that we can make all the entries zero by a series of moves.