<s>[INST] Find all real matrices such that $X^{3}-4X^{2}+5X=\begin{pmatrix} 10 & 20 \\ 5 & 10 \end{pmatrix}$ The following question come from the 1998 Romanian Mathematical Competition:

Find all matrices in $M_2(\mathbb R)$ such that $$X^{3}-4X^{2}+5X=\begin{pmatrix} 10 & 20 \\ 5 & 10 \end{pmatrix}$$

Can you guy please help me? Thanks a lot!
 [/INST] Let $X=\begin{pmatrix} a & b \cr c & d \end{pmatrix}$. Then the matrix equation immediately gives that $d=a$ and $b=4c$. Now it is easy to solve the last equations in $a$ and $c$. The only real solution is $a=2$ and $c=1$, and hence
$$
X=\begin{pmatrix} 2 & 4 \cr 1 & 2 \end{pmatrix}.
$$
For complex numbers there are several other solutions, e.g., 
$a=\frac{i + 6}{2},\; c= \frac{2- i }{4}$, or $a=-\frac{\sqrt{-5}}{2},\;c=-\frac{\sqrt{-5}}{4}$.
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