Exercise 5 — Candidates who have chosen the specialisation option Consider the matrix $A = \left( \begin{array} { l l } - 4 & 6 \\ - 3 & 5 \end{array} \right)$.
We call $I$ the identity matrix of order 2. Verify that $A ^ { 2 } = A + 2 I$.
Deduce an expression for $A ^ { 3 }$ and an expression for $A ^ { 4 }$ in the form $\alpha A + \beta I$ where $\alpha$ and $\beta$ are real numbers.
Consider the sequences $\left( r _ { n } \right)$ and $\left( s _ { n } \right)$ defined by $r _ { 0 } = 0$ and $s _ { 0 } = 1$ and, for all natural integer $n$, $$\left\{ \begin{array} { l }
r _ { n + 1 } = r _ { n } + s _ { n } \\
s _ { n + 1 } = 2 r _ { n }
\end{array} \right.$$ Prove that, for all natural integer $n , A ^ { n } = r _ { n } A + s _ { n } I$.
Prove that the sequence ( $k _ { n }$ ) defined for all natural integer $n$ by $k _ { n } = r _ { n } - s _ { n }$ is geometric with common ratio $- 1$. Deduce, for all natural integer $n$, an explicit expression for $k _ { n }$ as a function of $n$.
We admit that the sequence ( $t _ { n }$ ) defined for all natural integer $n$ by $t _ { n } = r _ { n } + \frac { ( - 1 ) ^ { n } } { 3 }$ is geometric with common ratio 2. Deduce, for all natural integer $n$, an explicit expression for $t _ { n }$ as a function of $n$.
From the previous questions, deduce, for all natural integer $n$, an explicit expression for $r _ { n }$ and $s _ { n }$ as a function of $n$.
Deduce then, for all natural integer $n$, an expression for $A ^ { n }$.
\textbf{Exercise 5 — Candidates who have chosen the specialisation option}
Consider the matrix $A = \left( \begin{array} { l l } - 4 & 6 \\ - 3 & 5 \end{array} \right)$.
\begin{enumerate}
\item We call $I$ the identity matrix of order 2.
Verify that $A ^ { 2 } = A + 2 I$.
\item Deduce an expression for $A ^ { 3 }$ and an expression for $A ^ { 4 }$ in the form $\alpha A + \beta I$ where $\alpha$ and $\beta$ are real numbers.
\item Consider the sequences $\left( r _ { n } \right)$ and $\left( s _ { n } \right)$ defined by $r _ { 0 } = 0$ and $s _ { 0 } = 1$ and, for all natural integer $n$,
$$\left\{ \begin{array} { l }
r _ { n + 1 } = r _ { n } + s _ { n } \\
s _ { n + 1 } = 2 r _ { n }
\end{array} \right.$$
Prove that, for all natural integer $n , A ^ { n } = r _ { n } A + s _ { n } I$.
\item Prove that the sequence ( $k _ { n }$ ) defined for all natural integer $n$ by $k _ { n } = r _ { n } - s _ { n }$ is geometric with common ratio $- 1$.\\
Deduce, for all natural integer $n$, an explicit expression for $k _ { n }$ as a function of $n$.
\item We admit that the sequence ( $t _ { n }$ ) defined for all natural integer $n$ by $t _ { n } = r _ { n } + \frac { ( - 1 ) ^ { n } } { 3 }$ is geometric with common ratio 2.\\
Deduce, for all natural integer $n$, an explicit expression for $t _ { n }$ as a function of $n$.
\item From the previous questions, deduce, for all natural integer $n$, an explicit expression for $r _ { n }$ and $s _ { n }$ as a function of $n$.
\item Deduce then, for all natural integer $n$, an expression for $A ^ { n }$.
\end{enumerate}