Let \(A=\left[\begin{matrix}1 & 2 \\ 1 & \alpha \end{matrix}\right]\) and \(B=\left[\begin{matrix}3 & 3 \\ \beta & 2\end…

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

[JEE Main 2026, 2 Apr (Shift 1)]