Mathematica: how to balance a matrix by proportional adjustment of rows and columns

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how to balance a matrix by proportional adjustment of rows and columns
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Thanks to all those great people for their contributions!

(mathematica.stackexchange.com/users/60365/tugrul-temel)Tugrul Temel
(mathematica.stackexchange.com/users/51/daniel-lichtblau)Daniel Lichtblau
(mathematica.stackexchange.com/users/72111/cvgmt)cvgmt
(mathematica.stackexchange.com/users/125/kglr)kglr
A special thanks goes out to the (https://mathematica.stackexchange.com/questions/288099/how-to-balance-a-matrix-by-proportional-adjustment-of-rows-and-columns)Stackexchange community

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