# Nonnegative Sums of Rows and Columns

Posted by Jason Polak on 11. December 2011 · Write a comment · Categories: elementary · Tags: , ,

For any $n\times n$ matrix $A$ with real entries, is it possible to make the sum of each row and each column nonnegative just by multiplying rows and columns by $-1$? In other words, you are allowed to multiply any row or column by $-1$ and repeat a finite number of times.

My fellow office mate Kirill, who also has a math blog, gave me this problem a few weeks ago and I thought about it for a few minutes here and there. The solution is in the fourth paragraph, so if you'd like to think about it yourself stop here before you get close.
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