\textbf{Exercise 2 — Candidates who have followed the specialization course}

We are given the matrices $M = \left( \begin{array} { c c c } 1 & 1 & 1 \\ 1 & - 1 & 1 \\ 4 & 2 & 1 \end{array} \right)$ and $I = \left( \begin{array} { l l l } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right)$.

\section*{Part A}
\begin{enumerate}
  \item Determine the matrix $M ^ { 2 }$. We are given $M ^ { 3 } = \left( \begin{array} { c c c } 20 & 10 & 11 \\ 12 & 2 & 9 \\ 42 & 20 & 21 \end{array} \right)$.
  \item Verify that $M ^ { 3 } = M ^ { 2 } + 8 M + 6 I$.
  \item Deduce that $M$ is invertible and that $M ^ { - 1 } = \frac { 1 } { 6 } \left( M ^ { 2 } - M - 8 I \right)$.
\end{enumerate}

\section*{Part B Study of a particular case}
We seek to determine three integers $a , b$ and $c$ such that the parabola with equation $y = a x ^ { 2 } + b x + c$ passes through the points $\mathrm { A } ( 1 ; 1 ) , \mathrm { B } ( - 1 ; - 1 )$ and $\mathrm { C } ( 2 ; 5 )$.
\begin{enumerate}
  \item Prove that the problem amounts to finding three integers $a , b$ and $c$ such that
$$M \left( \begin{array} { l } a \\ b \\ c \end{array} \right) = \left( \begin{array} { c } 1 \\ - 1 \\ 5 \end{array} \right)$$
  \item Calculate the numbers $a$, $b$ and $c$ and verify that these numbers are integers.
\end{enumerate}

\section*{Part C Return to the general case}
The numbers $a , b , c , p , q , r$ are integers. In a frame ( $\mathrm { O } , \vec { \imath } , \vec { \jmath }$ ), we consider the points $\mathrm { A } ( 1 ; p ) , \mathrm { B } ( - 1 ; q )$ and $\mathrm { C } ( 2 ; r )$. We seek values of $p , q$ and $r$ for which there exists a parabola with equation $y = a x ^ { 2 } + b x + c$ passing through A, B and C.
\begin{enumerate}
  \item Prove that if $\left( \begin{array} { l } a \\ b \\ c \end{array} \right) = M ^ { - 1 } \left( \begin{array} { c } p \\ q \\ r \end{array} \right)$ with $a , b$ and $c$ integers, then
$$\begin{cases} - 3 p + q + 2 r & \equiv 0 [ 6 ] \\ 3 p - 3 q & \equiv 0 [ 6 ] \\ 6 p + 2 q - 2 r & \equiv 0 [ 6 ] \end{cases}$$
  \item Deduce that $\left\{ \begin{array} { l l l } q - r & \equiv & 0 [ 3 ] \\ p - q & \equiv & 0 [ 2 ] \end{array} \right.$.
  \item Conversely, we admit that if $\left\{ \begin{array} { l } q - r \equiv 0 [ 3 ] \\ p - q \equiv 0 [ 2 ] \\ \mathrm { A } , \mathrm { B } , \mathrm { C } \text{ are not collinear} \end{array} \right.$ then there exist three integers $a , b$ and $c$ such that the parabola with equation $y = a x ^ { 2 } + b x + c$ passes through the points $\mathrm { A } , \mathrm { B }$ and C.\\
  a. Show that the points $\mathrm { A } , \mathrm { B }$ and C are collinear if and only if $2 r + q - 3 p = 0$.\\
  b. We choose $p = 7$. Determine integers $q , r , a , b$ and $c$ such that the parabola with equation $y = a x ^ { 2 } + b x + c$ passes through the points $\mathrm { A } , \mathrm { B }$ and C.
\end{enumerate}