Geodesic
There exist square real matrices in each dimension $n\ge2880$ which are not $DOTU$ matrices
Zapiski Nauchnykh Seminarov POMI, Studies in number theory. Part 7, Tome 106 (1981), pp. 134-136

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@article{ZNSL_1981_106_a7,
     author = {B. F. Skubenko},
     title = {There exist square real matrices in each dimension $n\ge2880$ which are not $DOTU$ matrices},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {134--136},
     publisher = {mathdoc},
     volume = {106},
     year = {1981},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1981_106_a7/}
}
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B. F. Skubenko. There exist square real matrices in each dimension $n\ge2880$ which are not $DOTU$ matrices. Zapiski Nauchnykh Seminarov POMI, Studies in number theory. Part 7, Tome 106 (1981), pp. 134-136. http://geodesic.mathdoc.fr/item/ZNSL_1981_106_a7/