\textbf{(For candidates who have followed the specialization course)}

As part of a study on social interactions between mice, researchers place laboratory mice in a cage with two compartments A and B. The door between these compartments is opened for ten minutes every day at noon.\\
We study the distribution of mice in the two compartments. It is estimated that each day:
\begin{itemize}
  \item $20\%$ of the mice present in compartment A before the door opens are found in compartment B after the door closes,
  \item $10\%$ of the mice that were in compartment B before the door opens are found in compartment A after the door closes.
\end{itemize}
We assume that initially, the two compartments A and B contain the same number of mice. We set $a_{0} = 0.5$ and $b_{0} = 0.5$.\\
For all natural integer $n$ greater than or equal to 1, we denote by $a_{n}$ and $b_{n}$ the proportions of mice present respectively in compartments A and B after $n$ days, after the door closes. We denote by $U_{n}$ the matrix $\binom{a_{n}}{b_{n}}$.

\begin{enumerate}
  \item Let $n$ be a natural integer.\\
  a. Justify that $U_{1} = \binom{0.45}{0.55}$.\\
  b. Express $a_{n+1}$ and $b_{n+1}$ as functions of $a_{n}$ and $b_{n}$.\\
  c. Deduce that $U_{n+1} = MU_{n}$ where $M$ is a matrix that we will specify.\\
  We admit without proof that $U_{n} = M^{n}U_{0}$.\\
  d. Determine the distribution of mice in compartments A and B after 3 days.
  \item Let the matrix $P = \left(\begin{array}{cc} 1 & 1 \\ 2 & -1 \end{array}\right)$.\\
  a. Calculate $P^{2}$. Deduce that $P$ is invertible and $P^{-1} = \frac{1}{3}P$.\\
  b. Verify that $P^{-1}MP$ is a diagonal matrix $D$ that we will specify.\\
  c. Prove that for any natural integer $n$ greater than or equal to 1, $M^{n} = PD^{n}P^{-1}$.\\
  Using computer algebra software, we obtain
  $$M^{n} = \left(\begin{array}{cc} \frac{1 + 2 \times 0.7^{n}}{3} & \frac{1 - 0.7^{n}}{3} \\ \frac{2 - 2 \times 0.7^{n}}{3} & \frac{2 + 0.7^{n}}{3} \end{array}\right).$$
  \item Using the help of the previous questions, what can we say about the long-term distribution of mice in compartments A and B of the cage?
\end{enumerate}