Let A be a 3 × 3 matrix such that X^T AX = O for all nonzero 3 × 1 matrices X = [x; y; z]. If A [1; 1; 1] = [1; 4; -5], A [1; 2; 1] = [0; 4; -8], and…

Let A be a $\displaystyle 3 \times 3$ matrix such that $\displaystyle \mathrm{X}^{\mathrm{T}} \mathrm{AX}=\mathrm{O}$ for all nonzero $\displaystyle 3 \times 1$ matrices $\displaystyle \mathrm{X}=\left[\begin{array}{l}x \\ y \\ z\end{array}\right]$. If $\displaystyle A\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=\left[\begin{array}{c}1 \\ 4 \\ -5\end{array}\right], A\left[\begin{array}{l}1 \\ 2 \\ 1\end{array}\right]=\left[\begin{array}{c}0 \\ 4 \\ -8\end{array}\right]$, and $\displaystyle \operatorname{det}(\operatorname{adj}(2(A+I)))=2^\alpha 3^\beta 5^\gamma, \alpha, \beta, \gamma \in \mathbb{N}$, then $\displaystyle \alpha^2+\beta^2+\gamma^2$ is $\displaystyle \_\_\_\_$.