What can be Jordan forms of matrix with charcterstic polynomial $(x-3)^2 (x-2)^4$ and minimal polynomial $(x-3)(x-2)^2$ The following question was asked in my linear algebra quiz and I was unable to think which result I should use to solve it.

Question: Let A be a $6 \times 6$ matrix over $\mathbb{R}$ with chacterstic polynomial =$(x-3)^2 (x-2)^4$ and minimal polynomial = $(x-3)(x-2)^2$ .Then Jordan canonical form of A can be? I have to find all the matrix which are possible.

Algebraic multiplicity of 2 and 3 are 4,2 respectively and Geometric multiplicity of 2 and 3 are 2 and 1 respectively.
But i don't know which result should be used to move foreward. I calculated these in case they are useful.
Can you please shed some light on this?
 A: If a factor $(x-\lambda)^k$ appears in the characteristic polynomial, then $k$ is total size of all Jordan $\lambda$-blocks.
If a factor $(x-\lambda)^k$ appears in the minimal polynomial, then $k$ is the size of the largest Jordan $\lambda$-block.
In our case we get:

*

*size of the largest $3$-block is $1$ and the total size is $2$ so there are two $3$-blocks of size $1$

*size of the largest $2$-block is $2$ and the total size is $4$ we can have two $2$-blocks of size $2$ or one $2$-block of size $2$ and two $2$-blocks of size $1$
$$\begin{bmatrix}
3 & & & & & \\
& 3 & & & & \\
& & 2 & 1 & & \\
& & & 2 &  & \\
& & & & 2 & 1 \\
& & & & & 2 
\end{bmatrix}\quad\text{ or } \quad
\begin{bmatrix}
3 & & & & & \\
& 3 & & & & \\
& & 2 & 1 & & \\
& & & 2 &  & \\
& & & & 2 &  \\
& & & & & 2 
\end{bmatrix}$$
A: By assumption, $A$'s group of elementary divisors can be one of the following:

*

*$\lambda - 3, \lambda - 3, (\lambda - 2)^2, (\lambda - 2)^2$;

*$\lambda - 3, \lambda - 3, (\lambda - 2)^2, \lambda - 2, \lambda - 2$.

Therefore, up to row/column permutations, there are two possibilities of $A$'s Jordan form
$\newcommand{\diag}{\mathrm{diag}}$
\begin{align*}
& \diag(3, 3, J_{(2)}(2), J_{(2)}(2)), \\
& \diag(3, 3, J_{(2)}(2), 2, 2),
\end{align*}
where $J_{(m)}(\lambda)$ denotes the Jordan block of size $m$ associated with eigenvalue $2$.
