An interesting matrix problem

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

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

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