The system, $$ \dot{\mathbf{x}}(t)=\left[\begin{array}{ll} 1 & 1 \\ 0 & 2 \end{array}\right] \mathbf{x}(t)+\left[\begin{array}{l} 1 \\ 0 \end{array}\right] u $$ with the same control approach $$ A_{c l}=A-B K=\left[\begin{array}{ll} 1 & 1 \\ 0 & 2 \end{array}\right]-\left[\begin{array}{l} 1 \\ 0 \end{array}\right]\left[\begin{array}{ll} k_{1} & k_{2} \end{array}\right]=\left[\begin{array}{cc} 1-k_{1} & 1-k_{2} \\ 0 & 2 \end{array}\right] $$ so that $$ \operatorname{det}\left(s I-A_{c l}\right)=\left(s-1+k_{1}\right)(s-2)=0 $$ So the feedback control can modify the pole at $s=1$, but it cannot move the pole at $s=2$.
cannot be stabilized with full state feedback, and the determinant of its controllability matrix is 0. Hence it is not controllable either.
But can it still be stabilizable? - Maybe using something else than full state feedback.