Answer
See below
Work Step by Step
Given: $p(\lambda)=(4-\lambda)^2(-6-\lambda)$
Find the eigenvalues:
$\lambda_1=4$ with multiplicity of 2
and $\lambda_2=-6$ with multiplicity of 1
The possibilities for $\lambda_1=4$ with multiplicity of 2 are:
One block of $2 \times 2$ matrix and Two blocks $1\times 1$ matrices.
The possibility for $\lambda_1=-6$ with multiplicity of 1 are:
One block of $1 \times 1$ matrix.
The set S of matrices include:
$J_1=\begin{bmatrix}
4 & 1 & 0\\
0 & 4 & 0\\
0 & 0 & -6
\end{bmatrix}$
$J_2=\begin{bmatrix}
4 & 0 & 0\\
0 & 4 & 0\\
0 & 0 & -6
\end{bmatrix}$
