Find the Jordan canonical forms of all $9\times 9$ matrices over $\mathbb{C}$ with minimal polynomial $x^2(x-3)^3$.
My method: each factor of the minimal polynomial corresponds to a type of Jordan blocks with their maximal orders equal to the multiplicity of the factor. Hence all possible Jordan blocks are (List A)
\begin{bmatrix} 0 \end{bmatrix}
\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}
\begin{bmatrix} 3 \end{bmatrix}
\begin{bmatrix}3 & 1 \\ 0 & 3 \end{bmatrix}
\begin{bmatrix} 3 & 1 & 0 \\ 0 & 3 & 1 \\0 & 0 & 3 \end{bmatrix}
Hence all possible matrices are
\begin{bmatrix} 0 & 1 & &&&&&& \\ 0 & 0 &&&&&&&\\&&3 & 1 & 0 &&&&\\ &&0 & 3 & 1 &&&&\\&&0 & 0 & 3 &&&&\\ &&&&&B&&&\end{bmatrix}
where $B$ are $4\times 4$ matrices chosen arbitrarily as a combination of all possible blocks listed in List A:
\begin{bmatrix} 0 &&& \\ &0&&\\&&0&\\&&&0\end{bmatrix}
\begin{bmatrix} 0 &&& \\ &0&&\\&&0&\\&&&3\end{bmatrix}
\begin{bmatrix} 0 &&& \\ &0&&\\&&3&\\&&&3\end{bmatrix}
\begin{bmatrix} 0 &&& \\ &3&&\\&&3&\\&&&3\end{bmatrix}
\begin{bmatrix} 3 &&& \\ &3&&\\&&3&\\&&&3\end{bmatrix}
\begin{bmatrix} 0 &&& \\ &3 & 1 & 0 \\ &0 & 3 & 1 \\&0 & 0 & 3 \end{bmatrix}
\begin{bmatrix} 3&&& \\ &3 & 1 & 0 \\ &0 & 3 & 1 \\&0 & 0 & 3 \end{bmatrix}
\begin{bmatrix} 0 &1&& \\ &0 & & \\ & &0 & \\& & &0 \end{bmatrix}
\begin{bmatrix} 0 &1&& \\ &0 & & \\ & &0 & \\& & &3 \end{bmatrix}
\begin{bmatrix} 0 &1&& \\ &0 & & \\ & &3 & \\& & &3 \end{bmatrix}
\begin{bmatrix} 0 &1&& \\ &0 & & \\ & &3 & 1 \\& & &3 \end{bmatrix}
\begin{bmatrix} 0 &&& \\ &0 & & \\ & &3 & 1 \\& & &3 \end{bmatrix}
\begin{bmatrix} 0 &&& \\ &3 & & \\ & &3 & 1 \\& & &3 \end{bmatrix}
\begin{bmatrix} 3 &&& \\ &3 & & \\ & &3 & 1 \\& & &3 \end{bmatrix}
\begin{bmatrix} 3 &1&& \\ &3 & & \\ & &3 & 1 \\& & &3 \end{bmatrix}
\begin{bmatrix} 0 &1&& \\ &0 & & \\ & &0 & 1 \\& & &0 \end{bmatrix}.
However, some classmates said I need to consider the primary factor and invariant factor thus cut off some of the possibilities. I am quite confused...