System of linear equations with matrix power

Question

I have a matrix $A\in\mathbb{R}^{d\times d}$, $z_1,\dots,z_{n+1}\in\mathbb{R}^d$, satisfying the following system of linear equations: \begin{align*} \begin{cases} A^nz_{n+1}=0,\\ A^{n-1}z_{n+1}+A^nz_{n}=0,\\ \dots \dots\\ z_{n+1}+Az_{n}+\dots+A^nz_1=0. \end{cases} \end{align*} I would like to conclude that each term in the summands is $0$. I know that if $A$ is symmetric, then $\mathrm{ker}(A)=\mathrm{ker}(A^i)$, $\forall i\ge 1$, then the above equation implies that all terms are $0$. This is not true in general, if we take $n=1$ and $A=\begin{bmatrix}0 & 1 \\ 0 & 0\end{bmatrix}$ and $z_2=[1, 0]$, $z_1=[-1, 0]$. What conditions do I need to impose on $A$?