Prove that $$\exp\left(-\sum_{k=1}^{n-1}\frac{N^k}{k}\right)=I_n-N$$ for $\lambda \in \mathbb{C}$ and $$N=\left(\begin{array}{cccc}{0} & {\lambda} & {0} & {0} \\ {0} & {0} & {\ddots} & {0} \\ {0} & {0} & {\ddots} & {\lambda} \\ {0} & {0} & {0} & {0}\end{array}\right)_{n\times n}.$$
This is a special case of the question I asked yesterday, where I got (quite good) answers, but unfortunately all of them use knowledge that I cannot yet use. Hence, I tried to reduce the problem to the special case for $N$ above that I'm actually interested in.
I am familiar with basic properties of the exponential function, but I have not seen any properties of other series defined on matrices. Any hints are welcome.
Let me know if followup-questions like those are not allowed.
