Nilpotent matrices and nilpotent level I would like to know what is a nilpotent matrix and nilpotency level? Are these matrices invertible? 

A matrix $A$ is nilpotent iff some power of it is zero, that is if there is a $k \in \mathbb N$ with $A^k = 0$. The smallest such $k$ is called the nilpotency degree of $A$ (you call this level of nilpotency?). 
If $A$ were nilpotent and invertible, we would have for $k$ with $A^k=0$ that
\[ \mathrm{Id} = (AA^{-1})^k = A^kA^{-k} = 0 \]
which is absurd.
