Let A be a 3 × 3 matrix such that X^T A X=O for all nonzero 3 × 1 matrices X= [ … ]. If A [ … ], and det( adj(2( A+ 1)))-2^α 3^β 5^γ, α, β, γ ∈ N,…

$\begin{aligned}
& X^{\top} A X=0 \
& (x y z)\left[\begin{array}{lll}
a_1 & a_2 & a_3 \
b_1 & b_2 & b_3 \
c_1 & c_2 & c_3
\end{array}\right]\left[\begin{array}{l}
x \
y \
z
\end{array}\right]=0 \
& (x y z)\left[\begin{array}{l}
a_1 x+a_2 y+a_3 z \
b_1 x+b_2 y+b_3 z \
c_1 x+c_2 y+c_3 z
\end{array}\right]=0 \
& x\left(a_1 x+a_2 y+a_3 z\right)+y\left(b_1 x+b_2 y+b_3 z\right) \
& +z\left(c_1 x+c_2 y+c_3 z\right) \
& a_1=0, b_2=0, c_3=0 \
& a_2+b_1=0, a_3+c_1=0, b_3+c_2=0 \
& A=\text { skew symmetric matrix } \
& A=\left[\begin{array}{ccc}
0 & x & y \
-x & 0 & z \
-y & -z & 0
\end{array}\right] ; A\left[\begin{array}{l}
1 \
1 \
1
\end{array}\right]=\left[\begin{array}{c}
1 \
4 \
-5
\end{array}\right] \
& \Rightarrow\left[\begin{array}{ccc}
0 & x & y \
-x & 0 & z \
-y & -z & 0
\end{array}\right]\left[\begin{array}{l}
1 \
1 \
1
\end{array}\right]=\left[\begin{array}{c}
1 \
4 \
-5
\end{array}\right] \
& x+y=1 \
& -x+z=4 \
& y+z=5 \
& {\left[\begin{array}{ccc}
0 & x & y \
-x & 0 & z \
-y & -z & 0
\end{array}\right]\left[\begin{array}{l}
1 \
2 \
1
\end{array}\right]=\left[\begin{array}{c}
1 \
4 \
-8
\end{array}\right]} \
& 2 x+y=0 \quad x=-1 \
& -x+z=4 \quad y=2 \
& -y-2 z=-8 \quad z=3
\end{aligned}$$\begin{aligned} & A=\left[\begin{array}{ccc}0 & -1 & 2 \ 1 & 0 & 3 \ -2 & -3 & 0\end{array}\right] \ & 2(A+l)=\left[\begin{array}{ccc}2 & -2 & 4 \ 2 & 2 & 6 \ -2 & -6 & 2\end{array}\right] \ & 2(A+l)=120 \ & \Rightarrow \operatorname{det}(\operatorname{adj}(2 A+l)) \ & \quad=120^2=2^6 \cdot 3^2 \cdot 5^2 \ & \therefore \quad \alpha=6, \beta=2, \gamma=2 \ & \text { Hence } \alpha^2+\beta^2+\gamma^2=6^2+2^2+2^2=44\end{aligned}$