$\begin{aligned}
& \mathrm{BCB}^{-1}=\mathrm{A} \
& \Rightarrow\left(\mathrm{BCB}^{-1}\right)\left(\mathrm{BCB}^{-1}\right)=\mathrm{A} \cdot \mathrm{A} \
& \Rightarrow \mathrm{BCI} \mathrm{CB}^{-1}=\mathrm{A}^2 \
& \Rightarrow \mathrm{BC}^2 \mathrm{B}^{-1}=\mathrm{A}^2 \
& \Rightarrow \mathrm{B}^{-1}\left(\mathrm{BC}^2 \mathrm{B}^{-1}\right) \mathrm{B}=\mathrm{B}^{-1} \text { (A.A)B }
\end{aligned}$
From equation (1) $\begin{aligned}
& \mathrm{C}^2=\mathrm{A}^{-1} \cdot \mathrm{A} \cdot \mathrm{B} \
& \mathrm{C}^2=\mathrm{B}
\end{aligned}\begin{aligned}
& \text { Also } \mathrm{AB}^{-1}=\mathrm{A}^{-1} \
& \Rightarrow \mathrm{AB}^{-1} \cdot \mathrm{A}=\mathrm{A}^{-1} \mathrm{A}=\mathrm{I} \
& \Rightarrow \mathrm{A}^{-1}\left(\mathrm{AB}^{-1} \mathrm{A}\right)=\mathrm{A}^{-1} \mathrm{I} \
& \mathrm{B}^{-1} \mathrm{~A}=\mathrm{A}^{-1}
\end{aligned}$
Now characteristics equation of C2 is $\begin{aligned}
& \left|C_2-\lambda I\right|=0 \
& |B-\lambda I|=0
\end{aligned}\begin{aligned} & \Rightarrow\left|\begin{array}{cc}1-\lambda & 3 \ 1 & 5-\lambda\end{array}\right|=0 \ & \Rightarrow(1-\lambda)(5-1)-3=0 \Rightarrow\left(\lambda^2-6 \lambda+5\right)-3=0 \ & \Rightarrow \lambda^2-6 \lambda+2=0 \ & \Rightarrow \beta^2-6 \mathrm{~B}+2 \mathrm{I}=0 \ & \Rightarrow \mathrm{C}^4-6 \mathrm{C}^2+2 \mathrm{I}=0 \ & \alpha=-6 \ & \beta=2 \ & \therefore 2 \beta-\alpha=4+6=10\end{aligned}$