\section*{Exercise 4 (For candidates who have followed the specialty course)}
We denote $\mathbb{Z}$ the set of integers.\\
In this exercise, we study the set $S$ of matrices that can be written in the form $A = \left(\begin{array}{cc} a & b \\ c & d \end{array}\right)$, where $a, b, c$ and $d$ belong to the set $\mathbb{Z}$ and satisfy: $ad - bc = 1$.\\
We denote $I$ the identity matrix $I = \left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right)$.

\section*{Part A}
\begin{enumerate}
  \item Verify that the matrix $A = \left(\begin{array}{rr} 6 & 5 \\ -5 & -4 \end{array}\right)$ belongs to the set $S$.
  \item Show that there exist exactly four matrices of the form $A = \left(\begin{array}{ll} a & 2 \\ 3 & d \end{array}\right)$ belonging to the set $S$; state them explicitly.
  \item a. Solve in $\mathbb{Z}$ the equation $(E): 5x - 2y = 1$. We may note that the pair $(1; 2)$ is a particular solution of this equation.\\
  b. Deduce that there exist infinitely many matrices of the form $A = \left(\begin{array}{cc} a & b \\ 2 & 5 \end{array}\right)$ that belong to the set $S$. Describe these matrices.
\end{enumerate}

\section*{Part B}
In this part, we denote $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ a matrix belonging to the set $S$. We recall that $a$, $b$, $c$ and $d$ are integers such that $ad - bc = 1$.

\begin{enumerate}
  \item Show that the integers $a$ and $b$ are coprime.
  \item Let $B$ be the matrix: $B = \left(\begin{array}{rr} d & -b \\ -c & a \end{array}\right)$\\
  a. Calculate the product $AB$. It is admitted that $AB = BA$.\\
  b. Deduce that the matrix $A$ is invertible and give its inverse matrix $A^{-1}$.\\
  c. Show that $A^{-1}$ belongs to the set $S$.
  \item Let $x$ and $y$ be two integers. We denote $x'$ and $y'$ the integers such that $\binom{x'}{y'} = A\binom{x}{y}$.\\
  a. Show that $x = dx' - by'$. It is admitted that likewise $y = ay' - cx'$.\\
  b. We denote $D$ the GCD of $x$ and $y$ and we denote $D'$ the GCD of $x'$ and $y'$. Show that $D = D'$.
  \item We consider the sequences of natural numbers $(x_n)$ and $(y_n)$ defined by: $x_0 = 2019$, $y_0 = 673$ and for any natural number $n$:
  $$\left\{\begin{array}{l} x_{n+1} = 2x_n + 3y_n \\ y_{n+1} = x_n + 2y_n \end{array}\right.$$
  Using the previous question, determine, for any natural number $n$, the GCD of the integers $x_n$ and $y_n$.
\end{enumerate}