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I've revised the question to describe my constraints more accuratly
Student88
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Binary cube root of a non-negative, integer square matrix with given diagonal

How can I find all binary matrices $A$ such that $A^3$ is a non-negative, integer square matrix and

$$diag\left(A^3\right)=b$$

for some given vector $b$? Is there a way to characterize all the solutions?

Student88
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