Tricky Dot Products (2011 Putnam A4)

There are n rows; the same row evens out with itself; the two different rows force odd one out. Is this possible for every n? Also, does this more or less require linear algebra, or is it more ad hoc? Let's find out!

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Reference:
After successfully solving the problem, I consulted the corresponding solution in Putnam Archive (https://kskedlaya.org/putnam-archive/).

Видео Tricky Dot Products (2011 Putnam A4) канала LetsSolveMathProblems