[QUE/VS-06002]

Find the matrix representation of the  operators \(A\) on the vector space of all  \(2\times 2\) real matrices, where
\[ A \begin{pmatrix} a & b \\c & d\end{pmatrix} =
    \begin{pmatrix}1 & 1\\1& 1 \end{pmatrix}
    \begin{pmatrix} a & b \\c & d\end{pmatrix},\]
in the basis
 \begin{equation*}
  e_1 = \begin{pmatrix}
          1 & 0\\0& 0&0
        \end{pmatrix};\quad
  e_2 = \begin{pmatrix}
         0 & 1 \\ 0 & 0
        \end{pmatrix};\quad
  e_3 = \begin{pmatrix}
         0 & 0 \\ 1 & 0
         \end{pmatrix};\quad
  e_4 = \begin{pmatrix}
          0 & 0 \\
          0 & 1
        \end{pmatrix}.
\end{equation*}