Let $F(\lambda)=I_p+C\left(\lambda I_n-A\right)^{-1} B$ and $G(\lambda)=I_p-C_1\left(\lambda I_n-\right.$ $\left.A_1\right)^{-1} B_1$. Show that if $C_1=C$ and $(C, A)$ is an observable pair, then $F(\lambda) G(\lambda)=I_p$ if and only if $B_1=B$ and $A_1=A-B C$.