An interesting matrix problem

Revision en10, by anothermousey, 2023-01-24 18:10:52

I was wondering a problem: Is it possible to construct a $$$n \times n$$$ matrix (call it A) such that each number from $$$1$$$ to $$$n$$$ appears exactly $$$n$$$ times and the result of $$$A \times A$$$ (matrix multiplication) is a zero matrix (after modulo $$$n \times n$$$ terms in the matrix by $$$n$$$). I know it is obvious for odd $$$n$$$ but how about even $$$n$$$?

Upd: After roughly 2 weeks the solution is done by Phd Ben Grossmann and Litho on math stackexchange. You can find the solution and proof here. Thanks div4only for providing me with great support. I will state the construction here only.

Satisfied matrix construction

I do not know if this contributes much to competitive programming, but it is fun doing these types of math problems.

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  Rev. Lang. By When Δ Comment
en10 English anothermousey 2023-01-24 18:10:52 8
en9 English anothermousey 2023-01-24 18:10:13 0 (published)
en8 English anothermousey 2023-01-24 18:09:59 2 (saved to drafts)
en7 English anothermousey 2023-01-24 16:50:28 0 (published)
en6 English anothermousey 2023-01-24 16:50:09 2
en5 English anothermousey 2023-01-24 14:42:32 1555 (saved to drafts)
en4 English anothermousey 2023-01-12 12:14:08 24 Tiny change: 'lt of A*A is a zero' -> 'lt of A*A (matrix multiplication) is a zero'
en3 English anothermousey 2023-01-12 12:09:40 4 Tiny change: 'mber from 0 to n-1 appears e' -> 'mber from 1 to n appears e'
en2 English anothermousey 2023-01-12 12:00:32 44
en1 English anothermousey 2023-01-12 11:24:31 261 Initial revision (published)