Generalisation of $A^{k}$ matrix

Question

Question: If a $n$-by-$n$ matrix $A$ be

$A=\begin{align} \begin{bmatrix} 0 & 1 & 0 & \dots & 0 & 0 \\ 0 & 0 & 1 & \dots & 0 & 0 \\ \vdots & \vdots &\vdots &&\vdots& \vdots\\ 0 & 0 & 0 & \dots & 0 & 1 \\ 1 & 0 & 0 & \dots & 0 & 0 \end{bmatrix} \end{align}$

Show that $A^{k}=\begin{bmatrix} O & I_{n-k}\\I_{k} & O\end{bmatrix}$, where $k=1, 2, \cdots, n$.

My attempt is to prove with mathematical induction, but it doesn't work as $I_{k}$ will expand its row and column from $1$ to $n$, leads the multiplication of partition matrices will not hold true.

I know this is a partition matrix with two zero matrices on the first row and column, while another two identity matrices on the sub-diagonals , and what I conclude for the proof is: multiply one-by-one, but the proof will not that strict and does not meet the expectation for generalisation.

Answer 1

Hint: Let $e_1,\dots, e_n$ be the standard basis. Then for any matrix $M$ (with $n$ columns), we have $Me_i$ is the $i$th column of $M$.
So that, $Ae_i=e_{i-1}$ with cyclic indexing.

$A^2e_i=Ae_{i-1}=e_{i-2}$, and so on..