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?

Two matrices $A$ and $B$ are similar if there exists a matrix $P$ such that $A=PBP^{-1}$.
The solutions to your equation are $x=2,3$. Thus, all matrices which satisfy your equation must be similar to $B=\begin{bmatrix}v_1&0\\0&v_2\end{bmatrix}$, where $v_1$ and $v_2$ are either $2$ or $3$.
Choosing $P=\begin{bmatrix}a&b\\c&d\end{bmatrix}$, all solutions to your equation are
$$
A=PBP^{-1}=\frac{1}{ad-bc}\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}v_1&0\\0&v_2\end{bmatrix}\begin{bmatrix}d&-b\\-c&a\end{bmatrix},
$$
for any choice of $a,b,c,d$ where $ad-bc\neq0$.
