Properties of Nilpotent Matrices I have $2$ questions concerning nilpotent matrices and commutativity 


  $\bullet$ If $M$ is an invertible matrix and $N$ is nilpotent, is then $M-N$ invertible ? 


When $M$ and $N$ are commute yes, the inverse is then $(M^{k-1}+M^{k-2}N+\dots N^{k-1})M^{-k}$ if $N$ has nilpotent index $k$, but if they do not commute this is not the inverse, but there might exist one, or not ?


  $\bullet$ Is the sum of $2$ nilpotent matrices again nilpotent ? 


Of course under the condition that they don't commute, otherwise one can use binomial formula. I only know that the product of nilpotent matrices can be non-nilpotent

can you help ?
No the the first question
$$
\begin{bmatrix}0&amp;1\\1&amp;0\end{bmatrix}
-
\begin{bmatrix}0&amp;1\\0&amp;0\end{bmatrix}
=
\begin{bmatrix}0&amp;0\\1&amp;0\end{bmatrix}.$$

No to the second question:
$$
\begin{bmatrix}0&amp;1\\0&amp;0\end{bmatrix}
+
\begin{bmatrix}0&amp;0\\1&amp;0\end{bmatrix}
=
\begin{bmatrix}0&amp;1\\1&amp;0\end{bmatrix}
$$

As you see, it is really the same example, with
$$
M = \begin{bmatrix}0&amp;1\\1&amp;0\end{bmatrix}
$$
invertible, and
$$
N = \begin{bmatrix}0&amp;1\\0&amp;0\end{bmatrix},
\qquad
M - N = \begin{bmatrix}0&amp;0\\1&amp;0\end{bmatrix}
$$
nilpotent.
If M is invertible matrix of order n then the characteristic polynomial of M is $$C_M(x)=(x-a_1)^{m_1}(x-a_2)^{m_2}(x-a_3)^{m_3}... (x-a_n)^m_n$$ where $a_{i's}$ are eigenvalues and are non zero since $M$ is invertible and $\sum_{i=1}^n m_i=n$ and the characteristic polynomial of N is $$C_N(x)=x^n$$
Now the characteristic polynomial of $M-N$ is 
$$C_{M-N}=(x-(a_1+0))^{m_1}(x-(a_2+0))^{m_2}......(x-(a_n+0))^{m_n}$$
Thus we can see that none of the eigenvalue is zero hence $M-N$ is invertible.
For the case of M and N both nilpotent the characteristic polynomial of M+N is 
$$C_{M+N}(x)=x^n $$
Thus all the eigenvalues are zero and thus $M+N$ is also nilpotent if both $M$ and $N$ are nilpotent