Find all matrices $A$ of order $2 \times 2$ that satisfy the equation $A^2-5A+6I = O$ 
Find all matrices $A$ of order $2 \times 2$ that satisfy the equation
$$
A^2-5A+6I = O
$$

My Attempt:
We can separate the $A$ term of the given equality:
$$
\begin{align}
A^2-5A+6I &= O\\
A^2-3A-2A+6I^2 &= O
\end{align}
$$
This implies that $A\in\{3I,2I\} = \left\{\begin{pmatrix}
3 & 0\\ 
0 & 3
\end{pmatrix},
\begin{pmatrix}
2 & 0\\ 
0 & 2
\end{pmatrix}\right\}$.
Are these the only two possible values for $A$, or are there other solutions?If there are other solutions, how can I find them?

$A^2 - 5A + 6 = 0$ is equivalent to $(A-2)(A-3) = 0$, which is equivalent to $Sp(A) \subset \{2, 3\}$.
Three cases are possible :


*

*$Sp(A) = \{2\}$, i.e. $A = 2I$

*$Sp(A) = \{3\}$, i.e. $A = 3I$

*$Sp(A) = \{2, 3\}$, i.e. $A$ is similar to $\begin{pmatrix}
2 & 0\\ 
0 & 3
\end{pmatrix}$

