Answer in the empty space below. Your answer should be carefully justified, and all the steps of your argument should be discussed in details. Leave the check-boxes empty, they are used for the grading.
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% Question A - 6points
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\begin{description}
\item[Question~\theAMCquestionaff :] \textit{This question is worth 6 points.}
\end{description}
\correctorSix{OPEN-A}{~}
\correctorStop
Let $\alpha\in \C$.
\begin{itemize}
\item [1.] Find the explicit formula for the elements of the matrix $A_n = \begin{pmatrix}\alpha &1\\ 0 &\alpha\end{pmatrix}^{n}$, where $n \geq 1$ is an integer. Show the formula by reccurance.
\item [2.] Let $\alpha = 1+\textrm{i}$. Calculate $\alpha^{99}$ and $\alpha^{100}$.
\item[Question~\theAMCquestionaff :] \textit{This question is worth 10 points.}
\end{description}
\correctorTenHalf{OPEN-B}{~}
\correctorStop
Let $\alpha\in \C$.
\begin{itemize}
\item [1.] Find the explicit formula for the elements of the matrix $A_n = \begin{pmatrix}\alpha &1\\ 0 &\alpha\end{pmatrix}^{n}$, where $n \geq 1$ is an integer. Show the formula by reccurance.
\item [2.] Let $\alpha = 1+\textrm{i}$. Calculate $\alpha^{99}$ and $\alpha^{100}$.
\item[Question~\theAMCquestionaff :] \textit{This question is worth 20 points.}
\end{description}
\correctorTwenty{OPEN-C}{~}
\correctorStop
Let $\alpha\in \C$.
\begin{itemize}
\item [1.] Find the explicit formula for the elements of the matrix $A_n = \begin{pmatrix}\alpha &1\\ 0 &\alpha\end{pmatrix}^{n}$, where $n \geq 1$ is an integer. Show the formula by reccurance.
\item [2.] Let $\alpha = 1+\textrm{i}$. Calculate $\alpha^{99}$ and $\alpha^{100}$.