Mathematica: Select unique row and column entry from an nxn matrix

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Select unique row and column entry from an nxn matrix
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(mathematica.stackexchange.com/users/69429/steve237)Steve237
(mathematica.stackexchange.com/users/62063/a-kato)A. Kato
(mathematica.stackexchange.com/users/58868/vindobona)vindobona
(mathematica.stackexchange.com/users/106/user1066)user1066
(mathematica.stackexchange.com/users/26598/roman)Roman
(mathematica.stackexchange.com/users/72953/ydd)ydd
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