<s>[INST] Eigenvalues and eigenvectors of $4\times 4$ matrix Let M the $4\times 4$ matrix: 
$M = \begin{pmatrix}
0 &-2  & 4 &-2 \\ 
1 &1  &-2  &-1 \\ 
0 &0  &0  &0 \\ 
1 &-1  &2  &-3 
\end{pmatrix}  $  and $M^n = (-1)^n2^{n-2}M^2$
 I determined 
$$e^M = I + M + M^2 \times \sum_{n=2}^{\infty}\frac{(-1)^n \times 2^n}{n!} $$
but can i use $e^M $ to determine Eigenvalues and eigenvectors of this matrix ? 
 [/INST] As @5xum already said: 4x4 is not very large. Furthermore you can notice the row of zeros (which is really nice).

$$\det{(M-I\lambda)} = \det\begin{pmatrix}
0-\lambda &-2  & \color{red}4 &-2 \\ 
1 &1 - \lambda  &\color{red}{-2}  &-1 \\ 
\color{red}0 &\color{red}0  &\color{red}{-\lambda}  &\color{red}0 \\ 
1 &-1  &\color{red}2  &-3-\lambda 
\end{pmatrix}  $$

$$
=-\lambda \cdot \det\begin{pmatrix}
0-\lambda &-2 &-2 \\ 
1 &1 - \lambda &-1 \\ 
1 &-1  &-3-\lambda 
\end{pmatrix}  $$

This can be easily calculated:
$$ = -\lambda \cdot \left( -\lambda \cdot (1 - \lambda) \cdot (-3-\lambda) + (-2) \cdot (- 1) \cdot  1 +  (-2) \cdot 1 \cdot (-1)\right ....)$$
</s>