$$ \begin{cases} \dot{x}=\begin{pmatrix} -2 & 0 \\ 0 & -2 \\ \end{pmatrix} x+\begin{pmatrix} 2 \\ 2 \\ \end{pmatrix} u \\ y=Cx \end{cases} $$
How can I show that if $p_1 \neq -2$ and $p_2 \neq -2$, then it is impossible to apply pole placement for the above system by using $$det(sI-A-BK)=0.$$
After taking the determination I got the following equation. $$(s+2)(s+2-2k_1-2k_2)=0$$
Where $K=(k_1,k_2)$
You can also notice that the system is not controllable. Controllability is necessary as well as sufficient for placement of poles in arbitrary locations.