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?

As in general we have

$$
\mathbf{A}^2 - \big( \lambda_1 + \lambda_2 \big) \mathbf{A}
+ \lambda_1 \lambda_2 \mathbf{I} = 0, \tag 1
$$

we see that in this case

$$
\lambda_1 = 2, \quad \lambda_2 = 3. \tag 2
$$

So the general solution is given by

$$
\bbox[16px,border:2px solid #800000]
 { \mathbf{B} \pmatrix{ 3 & 0 \\ 0 & 2} \mathbf{B}^{-1}, } \tag 3
$$

for any matrix such that $\det(\mathbf{B}) \ne 0$.
Note that

$$
\pmatrix{0 & 1 \\ 1 & 0} \pmatrix{3 & 0 \\ 0 & 2} \pmatrix{0 & 1 \\ 1 & 0}
= \pmatrix{ 2 & 0 \\ 0 & 3}. \tag 4
$$

