Problem of March 03 2019

Problem. Let $A$ be a $n\times n$ complex matrix such that $(A’)^m=A^k$, where $A’$ stands for the transpose of $A$ and $m,k$ are distinct positive integers. Prove that the eigenvalues of $A$ are zero or root of unity ( $\xi^\ell =1$ for some positive integer $\ell$).

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