Answer
See below
Work Step by Step
The matrices that have five Jordan blocks corresponds to five linearly independent eigenvectors of $A$.
The set S of matrices include:
$\begin{bmatrix}
6 & 1 & 0 & 0 & 0 & 0 & 0\\
0 & 6 & 1 & 0 & 0 & 0 & 0\\
0 & 0 & 6 & 0& 0 & 0 & 0\\
0 & 0 & 0 & 6 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 6 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & -2 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & -2
\end{bmatrix}$
$\begin{bmatrix}
6 & 1 & 0 & 0 & 0 & 0 & 0\\
0 & 6 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 6 & 1 & 0 & 0 & 0\\
0 & 0 & 0 & 6 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 6 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & -2 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & -2
\end{bmatrix}$
$\begin{bmatrix}
6 & 1 & 0 & 0 & 0 & 0 & 0\\
0 & 6 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 6 & 0& 0 & 0 & 0\\
0 & 0 & 0 & 6 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 6 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & -2 & 1\\
0 & 0 & 0 & 0 & 0 & 0 & -2
\end{bmatrix}$
