$A^n = B^n$ does not imply $A = B$ for matrices

Matrix powers are not injective: $A^2=(-A)^2$ can hold with $A\ne -A$.Suggest a slogan

For square matrices $A,B$ and $n\ge2$, $A^n=B^n$ does not imply $A=B$. In $\Mat_2(\RR)$, let \[ A=\begin{bmatrix}0&1\\-1&0\end{bmatrix}. \] Then \[ A^2=\begin{bmatrix}-1&0\\0&-1\end{bmatrix}=(-A)^2, \] but $A\ne-A$.

For matrices, does $A^n=B^n\implies A=B$?