Balancing matrices with line shifts

We give a purely deterministic proof of the following theorem of J. Komlós and M. Sulyok. LetA=(aij),aij=±1 be ann×n matrix. One can multiply some rows and columns by −1 such that the absolute value of the sum of the elements of the matrix is ≦2 ifn is even and 1 ifn is odd. Note that Komlós and Sulyok applied probabilistic ideas and so their method worked only forn>n0.