The exercise is to prove how many matrices $X$ are such that $X^m=A$, knowing that $A$ have all $3$ eingenvalues $0$ and that $A^2≠0$. For $m=1$ we have $X=A$, but for $m>1$, if what I want to prove is true, there are no solutions, but I couldn't to prove this result.

  • $\begingroup$ The characteristic polynomial has degree $n$, so it is $x^n$ $\endgroup$
    – Bernard
    Aug 22 '18 at 0:51
  • $\begingroup$ It follows from the Cayley-Hamilton theorem $\endgroup$ Aug 22 '18 at 1:08
  • $\begingroup$ See this answer $\endgroup$ Aug 22 '18 at 1:10

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