Nilpotent matrix/Exponential mapping/Properties/Exercise

Let ${}n\in \mathbb {N} _{+}$, and let a field ${}K$ of characteristic ${}0$ be fixed. For a nilpotent ${}n\times n$-matrix ${}M$, let ${}\exp M$ be defined by

${}\exp M=\sum _{k=0}^{n}{\frac {1}{k!}}M^{k}\,.$

a) Show that for commuting nilpotent matrices ${}M,N$, the equality

${}\exp {\left(M+N\right)}=\exp M\circ \exp N\,$

holds.

b) Show that for a nilpotent matrix ${}M$, the matrix ${}\exp M$ is invertible.

c) Show that for a nilpotent matrix ${}M$, the matrix ${}\exp M$ is unipotent.

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