Let $A\in M_n(K)$ be a matrix over algebraic closed field $K$, where $n>1.$
When $A^i=A^j$ for $i,j\geq 0$ such that $i\neq j$?
I tried solve it by Jordan form of matrix $A;$ is is sufficient to answer when $J^i=J^j,$ where $J$ is Jordan-form matrix and $i\neq j.$ I know formula for a power of Jordan blocks, but I had problem to make computations for non-diagonal entries of blocks.
I tried also solve it by minimal polynomial $A$ (here $A$ satisfy a polynomial $f(x)=x^i-x^j.$)
Could anyone help me work out this problem? (I would be grateful for a hint; I don't want a full solution.)