Find minimal polynomial of given matrix I want to find the minimal polynomial of this matrix
$$\left[\begin{array}{ccccc}
2 & 0 & 0 & 0 & 0\\
4 & 2 & 0 & 0 & 0\\
1 & 4 & 2 & 0 & 0\\
0 & 0 & 0 & 3 & 0\\
0 & 0 & 0 & 0 & 3
\end{array}\right]$$
I found it charastic polynomial $\chi_{A}=\left(2-x\right)^{3}\left(3-x\right)^{2}$
I also know that the minimal polynomial irreducible factors have the same irreducible factors as charastic polynomial so
$$m_{A}\left(x\right)\in\left\{ \left(2-x\right)^{3}\left(3-x\right)^{2},\left(2-x\right)^{3}\left(3-x\right),\left(2-x\right)^{2}\left(3-x\right),\left(2-x\right)\left(3-x\right),\left(2-x\right)\left(3-x\right)^{2}\right\}  $$
but I don't know how to continue from here

The minimal polynomial is $(X-2)^3(X-3)$. This a triangular matrix with a $2\times 2$ diagonal block whose eigenvalue is $3$ thus whose minimal polynomial is $X-3$. There exists a block which is a $3\times 3$ matrix which has $2$ on the diagonal. The minimal polynomial of this block is $(X-2)^3$.
