Let $n$ be a non-zero natural number.

\begin{enumerate}
  \item Let $p$ be a vector projection of rank $r \in \mathbb { N }$.
  \begin{enumerate}
    \item[1.1.] Give, as a function of $r$, a matrix $W$ of $p$ in an adapted basis.
    \item[1.2.] Give the possible spectra of $W$.
    \item[1.3.] Compare $\boldsymbol { \operatorname { rg } } ( W )$ and $\boldsymbol { \operatorname { tr } } ( W )$.
    \item[1.4.] Calculate $\boldsymbol { \operatorname { det } } ( W )$.
  \end{enumerate}
\end{enumerate}

We consider the family $X _ { 1 } , \ldots , X _ { n }$ of independent random variables defined on the same probability space $( \Omega , \mathscr { A } , \mathbb { P } )$ all following the Bernoulli distribution with parameter $p \in ]0,1[$.

Let $M$ be a discrete random variable from $\Omega$ to $\mathscr { M } _ { n } ( \mathbb { R } )$ such that for all $\omega$ in $\Omega , M ( \omega )$ is diagonalisable and similar to $\Delta ( \omega ) = \operatorname { diag } \left( X _ { 1 } ( \omega ) , \ldots , X _ { n } ( \omega ) \right)$.

\begin{enumerate}
  \setcounter{enumi}{1}
  \item We denote by $T$ the random variable $\mathbf { tr } ( M )$.
  \begin{enumerate}
    \item[2.1.] Determine $T ( \Omega )$, that is the set of values taken by the random variable $T$.
    \item[2.2.] Give the probability distribution of $T$ and the expectation of the random variable $T$.
  \end{enumerate}
  \item Deduce the probability distribution of the random variable $R = \mathbf { rg } ( M )$.
  \item We denote by $D$ the random variable $\boldsymbol { \operatorname { det } } ( M )$.
  \begin{enumerate}
    \item[4.1.] Determine $D ( \Omega )$.
    \item[4.2.] Give the probability distribution of $D$ and calculate the expectation of the random variable $D$.
  \end{enumerate}
  \item We propose to determine the probability of the event $Z$: ``the eigenspaces of the matrix $M$ all have the same dimension''.
  \begin{enumerate}
    \item[5.1.] We denote by $V$ the event: ``$M$ has only one eigenvalue''. Calculate $\mathbb { P } ( V )$.
    \item[5.2.] Suppose $n$ is odd. Determine $\mathbb { P } ( Z )$.
    \item[5.3.] Suppose $n$ is even and set $n = 2 r$. Calculate $\mathbb { P } ( T = r )$. Deduce $\mathbb { P } ( Z )$.
  \end{enumerate}
  \item For all $\omega \in \Omega$, we denote $U ( \omega ) = \left( \begin{array} { c } X _ { 1 } ( \omega ) \\ \vdots \\ X _ { n } ( \omega ) \end{array} \right) \in \mathscr { M } _ { n , 1 } ( \mathbb { R } )$ and $A ( \omega ) = U ( \omega ) \times ( U ( \omega ) ) ^ { \top } = \left( a _ { i j } ( \omega ) \right) _ { ( i , j ) \in \llbracket 1 , n \rrbracket ^ { 2 } }$.
  \begin{enumerate}
    \item[6.1.] Let $\omega \in \Omega$. Determine, for all pairs $( i , j ) \in \llbracket 1 , n \rrbracket ^ { 2 } , a _ { i j } ( \omega )$.
    \item[6.2.] Give the probability distribution of each random variable $a _ { i j }$.
    \item[6.3.] Show that $\operatorname { tr } ( A ) = \sum _ { i = 1 } ^ { n } X _ { i }$.
    \item[6.4.] Determine the values taken by the random variable $\boldsymbol { \operatorname { rg } } ( A )$.
    \item[6.5.] For all $\omega$ in $\Omega$, give the eigenvalues of the matrix $A ( \omega )$.
    \item[6.6.] Determine the probability distribution of the random variable $\mathbf { rg } ( A )$.
  \end{enumerate}
\end{enumerate}