Let
$$ P = \begin{bmatrix} 0 & d & 0 & 0 & \ldots & 0 \\ 0 & r & d & 0 & \ldots & 0 \\ 0 & 0 & r & d & \ldots & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & d \\ 0 & 0 & 0 & 0 & \ldots & r \\ \end{bmatrix}$$
be a square $n \times n$ matrix. Give a general formula for the elements of $P^k$, where $k \in \mathbb{N}$ and analyze the asymptotic behavior of $P^k x$ for $k \to \infty $
Hint: $x_i^k \approx k^{n-i} \lambda^k c_i (x) $ for big $k$ where $ c_i (x) $ is constant.
I would be grateful for any hints.