Symmetric and skew-symmetric matricesHard

Properties of Skew Symmetric Matrices

Let be a 3 × 3 matrix such that X^T AX = O for all nonzero 3 × 1 matrices X = $\left[ \begin{matrix} x \hfill \cr y \hfill \cr z \hfill \cr \end{matrix} \right]$ .
If $A\left[ \begin{matrix} 1 \hfill \cr 1 \hfill \cr 1 \hfill \cr \end{matrix} \right] = \left[ \begin{matrix} 1 \hfill \cr 4 \hfill \cr - 5 \hfill \cr \end{matrix} \right],A\left[ \begin{matrix} 1 \hfill \cr 2 \hfill \cr 1 \hfill \cr \end{matrix} \right] = \left[ \begin{matrix} 0 \hfill \cr 4 \hfill \cr - 8 \hfill \cr \end{matrix} \right]$ , and det (adj(2(A + I))) = 2^α3^β5^γ, α, β, γ , ∈ N, then
α² + β² + γ² is

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