Let \(A\) be a \(3 \times 3\) matrix such that \({A}^{T}\left[\begin{matrix}1 \\ 0 \\ 1\end{matrix}\right]=\left[\begin{…

Let \(A\) be a \(3 \times 3\) matrix such that \({A}^{T}\left[\begin{matrix}1 \\ 0 \\ 1\end{matrix}\right]=\left[\begin{matrix}5 \\ 2 \\ 2\end{matrix}\right],{A}^{T}\left[\begin{matrix}0 \\ 0 \\ 1\end{matrix}\right]=\left[\begin{matrix}3 \\ 1 \\ 1\end{matrix}\right],A\left[\begin{matrix}1 \\ 0 \\ 1\end{matrix}\right]=\left[\begin{matrix}3 \\ 4 \\ 4\end{matrix}\right]\) and \(A\left[\begin{matrix}0 \\ 0 \\ 1\end{matrix}\right]=\left[\begin{matrix}1 \\ 3 \\ 1\end{matrix}\right]\). If \(\operatorname{det}(A)=1\), then \(\operatorname{det}\left(\operatorname{adj}\left(A^2+A\right)\right)\) is equal to:

[JEE Main 2026, 5 Apr (Shift 1)]