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Can a 0-1 matrix square to the matrix with all ones?

Source: IMC 2024, Problem 3

August 7, 2024
linear algebramatrixSquare matrix

Problem Statement

For which positive integers nn does there exist an n×nn \times n matrix AA whose entries are all in {0,1}\{0,1\}, such that A2A^2 is the matrix of all ones?
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