\textbf{Exercise 4 (Candidates who have followed the specialization course)}

Two column matrices $\binom { x } { y }$ and $\binom { x ^ { \prime } } { y ^ { \prime } }$ with integer coefficients are said to be congruent modulo 5 if and only if $\left\{ \begin{array} { l } x \equiv x ^ { \prime } [ 5 ] \\ y \equiv y ^ { \prime } [ 5 ] \end{array} \right.$.\\
Two square matrices of order $2 \left( \begin{array} { l l } a & c \\ b & d \end{array} \right)$ and $\left( \begin{array} { l l } a ^ { \prime } & c ^ { \prime } \\ b ^ { \prime } & d ^ { \prime } \end{array} \right)$ with integer coefficients are said to be congruent modulo 5 if and only if $\left\{ \begin{array} { l } a \equiv a ^ { \prime } [ 5 ] \\ b \equiv b ^ { \prime } [ 5 ] \\ c \equiv c ^ { \prime } [ 5 ] \\ d \equiv d ^ { \prime } [ 5 ] \end{array} \right.$.

Alice and Bob want to exchange messages using the procedure described below.
\begin{itemize}
  \item They choose a square matrix M of order 2, with integer coefficients.
  \item Their initial message is written in capital letters without accents.
  \item Each letter of this message is replaced by a column matrix $\binom { x } { y }$ deduced from the table below: $x$ is the digit located at the top of the column and $y$ is the digit located to the left of the row; for example, the letter T in an initial message corresponds to the column matrix $\binom { 4 } { 3 }$.
  \item We calculate a new matrix $\binom { x ^ { \prime } } { y ^ { \prime } }$ by multiplying $\binom { x } { y }$ on the left by the matrix $M$:
\end{itemize}
$$\binom { x ^ { \prime } } { y ^ { \prime } } = \mathrm { M } \binom { x } { y } .$$

\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
 & 0 & 1 & 2 & 3 & 4 \\
\hline
0 & A & B & C & D & E \\
\hline
1 & F & G & H & I & J \\
\hline
2 & K & L & M & N & O \\
\hline
3 & P & Q & R & S & T \\
\hline
4 & U & V & X & Y & Z \\
\hline
\end{tabular}
\end{center}