Let A= [1, 2; 1, α] and B= [3, 3; β, 2]. If A^2-4 A+I=O and B^2-5 B-6 I=O, then among the two statements: (S1): [( B - A )( B + A )]^T = [13, 15; 7,…

Let $\displaystyle A=\left[\begin{array}{ll}1 & 2 \\ 1 & \alpha\end{array}\right]$ and $\displaystyle B=\left[\begin{array}{ll}3 & 3 \\ \beta & 2\end{array}\right]$. If $\displaystyle A^2-4 A+I=O$ and $\displaystyle B^2-5 B-6 I=O$, then among the two statements : (S1): $\displaystyle [(\mathrm{B}-\mathrm{A})(\mathrm{B}+\mathrm{A})]^{\mathrm{T}}=\left[\begin{array}{cc}13 & 15 \\ 7 & 10\end{array}\right]$ and $$(\mathrm{S} 2): \operatorname{det}(\operatorname{adj}(\mathrm{A}+\mathrm{B}))=-5, $$