Problem definition
Given an integer $N>1$, let $A_N$ be the following $N\times N$ matrix \begin{equation*}A_N\triangleq \left[\begin{array}{c|c} & I_{N-1} \\ \hline 0_1 & \end{array}\right]\end{equation*} where $0_{a}$ is a $a\times a$ matrix with each entries equal to zero and $I_a$ is the $a\times a$ identity matrix. The north west block and the south east block are completed with zeros.
I'm interested in the explicit expression its exponential $F_N$ for any given $N$ \begin{equation*}F_N\triangleq \exp(A_N)\end{equation*}
Partial solution
Since $A_N$ is sparse, I think that the most straightforward way to get the answer consists in using the definition of matrix exponential \begin{equation*}\exp(A_N)\triangleq \sum_{k=0}^{\infty} \frac{A_N^k}{k!}\end{equation*} using this formula, the problem reduces to compute the powers of $A_N$.
I've noticed, by considering the special cases with $N\triangleq 2$ and $N\triangleq 3$, that as the exponent $k$ increase the identity block inside $A_N$ reduce its dimension until it vanishes. My conjecture, by denoting as $0_0$ and $I_0$ an empty block of numbers, is that \begin{equation*}A_N^k\triangleq \left[\begin{array}{c|c} & I_{N-k} \\ \hline 0_k & \end{array}\right] \qquad k=0,1,\dots,N\end{equation*} and $A_N^k=0_N$ for all $k>N$.
example 1: $N\triangleq2$
\begin{equation*} A_2^0\triangleq\left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right]=\left[\begin{array}{c|c} & I_{2} \\ \hline 0_0 & \end{array}\right]\end{equation*} \begin{equation*} A_2^1\triangleq\left[\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array}\right]=\left[\begin{array}{c|c} & I_1 \\ \hline 0_1 & \end{array}\right]\end{equation*} \begin{equation*} A_2^2=\left[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}\right]=\left[\begin{array}{c|c} & I_0 \\ \hline 0_2 & \end{array}\right]\end{equation*}
example 2: $N\triangleq3$
\begin{equation*} A_3^0\triangleq\left[\begin{array}{cc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]=\left[\begin{array}{c|c} & I_3 \\ \hline 0_0 & \end{array}\right]\end{equation*} \begin{equation*} A_3^1\triangleq\left[\begin{array}{ccc} 0 & 1 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0 \end{array}\right]=\left[\begin{array}{c|c} & I_2 \\ \hline 0_1 & \end{array}\right]\end{equation*} \begin{equation*} A_3^2=\left[\begin{array}{cc} 0 & 0 & 1\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array}\right]=\left[\begin{array}{c|c} & I_1 \\ \hline 0_2 & \end{array}\right]\end{equation*} \begin{equation*} A_3^3=\left[\begin{array}{cc} 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array}\right]=\left[\begin{array}{c|c} & I_0 \\ \hline 0_3 & \end{array}\right]\end{equation*}
Question
Is it true my conjecture? In that case, how can it be proven?