X,Y∊ℝᵐˣᵐ, X²+Y²=XY, det{XY‒YX}≠0 ⟹3|m && A∊{0,1}ᵐˣᵐ satisfies A² is all-one⟹m is a perfect square

Notes: https://drive.google.com/file/d/1wubH... Consider matrices X,Y ∊ ℝᵐˣᵐ that satisfy X² + Y² = XY Prove that: if XY‒YX is invertible, then m is divisible by 3 Matrix A ∈ ℝᵐˣᵐ is binary, i.e., all the elements of A are either 0 or 1. Moreover, A satisfies X²=Ωₘ, where Ωₘ is the m-by-m all-one matrix. For which values of m does such matrix A exists? I saw the second question with the name Alex Avdiushenko