Let $K$ be a commutative field and ${\rm M}_n (K)$ be the ring of $n\times n$ square matrices with coefficients in $K$ ($n\geqslant 1$ is an integer). For $k\geqslant 1$ and $A =(a_{ij})_{1\leqslant i,j\leqslant n}\in {\rm M}_n (K)$, define: $A^{[k]} =(a_{ij}^k )_{1\leqslant i,j\leqslant n}$.

Is the description of all matrices $A\in {\rm M}_n (K)$ satisfying $A^k =A^{[k]}$, for all $k\geqslant 1$, known? If yes do you have a reference ?