Let me explain it with 0,1 matrix.
0 0 1
0 1 1
1 0 1
1 1 1
1,3,5,7 changing to 1,11,101,111.the sum of every colum is 0.We call it even, First no matter how to try,can't keep even,but second can let it even or odd.If who take the last who win,the second always keep even,other case second only at last let it odd.
This method can use for any m*n Matrix.