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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