[BONUS PROBLEM (+20 pts) Solve the following problem without the use of a computer.
Please make sure to demonstrate all the steps of your work, as presenting only the final answer will
not earn any credit.
Consider the matrix
$A = \begin{bmatrix} \alpha & \beta & 1 \\ 0 & \alpha & \beta \\ 0 & 0 & \alpha \end{bmatrix}$, $\alpha, \beta \in \mathbb{R}$
a. Calculate the eigenvalues and eigenvectors, and, if necessary, the generalized eigenvectors.
b. Calculate the Jordan form using the obtained eigenvectors.
c. Calculate $A^n$, $n \in \mathbb{N}$. Hint: You must use the matrix function for the Jordan form.
d. Calculate $e^{At}$. Hint: You must use the matrix function for the Jordan form.
