Replace $-$ by 0 and $+$ by 1. Suppose the sum of elements in each column are different. Since these sum are between $0$ and $n$ and there are $n$ sum all added together leading to $n^2/2$, we must have only one missing number $k$ such that $(0+1+...+n)-k=n^2/2$. Hence the missing number is $n/2$. Now the previous step guarantees that there is a column with only 0 and a column with only 1. Now if you suppose furthermore that the sum of element is each row is also different, then similarly, you get a row with only 0. But this is contradiction as we cannot have a row with only 0 and a column with only 1 as the intersection will have 0 and 1 at the same time.