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