> 
> Consider $$X=\Big\{A \in M\_3(\Bbb R): \rho\_A(x)=x^3-3x^2+2x-1\Big\}$$ where $\rho\_A(x)$ is the characteristic polynomial of $A$ and $M\_3(\Bbb R)$ is the space of all $3 \times 3$ matrices over $\Bbb R$.
> 
> 
> Is $X$ compact in $M\_3(\Bbb R)$ ?
> 
> 
> 

---

My try: I confused with only the $2x$ term in $\rho\_A(x)$. Because , if the $2x$ term not appear , then $X$ becomes $$X=\{A \in M\_3(\Bbb R): \text{trace}(A)=3,\det A=1\} $$ which is unbounded, since $$(\forall n \in \Bbb N):\begin{pmatrix} 1&0 & n\\0&1&0\\0&0&1 \end{pmatrix} \in X$$

But here the problem is, the appearance of $2$. I Know $$2=A\_{11}+A\_{22}+A\_{33}$$

so I think in this case the set becomes bounded and closed

Any help?