Does exists a matrix $X$ for how many $n$ such that $X^n=A$?

Question

Let

$$ A= \begin{pmatrix} 0 & 1 & 2 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{pmatrix} $$

For how many $n$ is there a matrix $X$ such that $X^n=A$?

Answer 1

Hint: what could be the minimal polynomial of such an $X$?

Answer 2

For how many $n$? Only for finitely many $n$. More precisely only for $n\le 2$, since $X$ must be nilpotent because of $A^2=0$ and $X^n=A$. However, a nilpotent matrix $X\in M_3(K)$ satisfies $X^3=0$. It follows that $X^n=0\neq A$ for all $n\ge 3$.