Suppose $(A_1, B_1)$ and $(A_2, B_2)$ are both stabilizable. Then we know that we can find some $K_1$ and $K_2$ to make $A_1+B_1K_1$ and $A_2+B_2K_2$ Hurwitz, respectively. Now, for non-zero constant matrices $C_1$ and $C_2$ with proper dimensions, can we always design $K_1$ and $K_2$ to make the following matrix Hurwitz? If so, how to prove it? Thanks in advance!
$$ \left[ {\begin{array}{*{20}c} {A_1 + B_1 K_1 } & {C_1 } \\ {C_2 } & {A_2 + B_2 K_2 } \\ \end{array}} \right] $$