A variable matrix $V=\left(\begin{array}{ccc}x& y& z\\ y& z& x\\ z& x& y\end{array}\right)$ such that $VA=\left(\begin{array}{c}0\\ 0\\ 0\end{array}\right)$ where $A=\left(\begin{array}{c}\log a\\ \log b\\ \log c\end{array}\right)$ and a, b, c distinct +ve numbers in G.P. If (x, y, z) is solution of VA = 0 lie on line $\frac{x}{\alpha }=\frac{y}{\beta }=\frac{z}{\gamma }is(x,y,z\ne 0)$ then $\frac{\alpha }{\beta }+\frac{\beta }{\gamma }+\frac{\gamma }{\alpha }$

$VA=0$

$\left(\log a\right)x+\left(\log b\right)y+\left(\log c\right)z=0\Rightarrow \left(\log b\right)x+\left(\log c\right)y+\left(\log a\right)z=0$

$\left(\log c\right)x+\left(\log a\right)y+\left(\log b\right)z=0$ is homogeneous system of equation in x, y, z and $(x,y,z)\ne (0,0,0)$

$\Rightarrow \Delta =\left|\begin{array}{ccc}\log a& \log b& \log c\\ \log b& \log c& \log a\\ \log c& \log a& \log b\end{array}\right|=0\Rightarrow \log a+\log b+\log c=0\Rightarrow \log \left(abc\right)=0$

$abc=1\Rightarrow b^3=1(b^2=ac)$

$b=1ac=1c=\frac{1}{a}$

Equation becomes $\Rightarrow \left(\log a\right)x-\left(\log b\right)z=0\Rightarrow -\left(\log a\right)y+\left(\log a\right)z=0$

$-\left(\log a\right)x-\left(\log a\right)y=0$

$x-y=0\Rightarrow y-z=0\Rightarrow x-y=0$

$x=y=z\Rightarrow \frac{x}{1}=\frac{y}{1}=\frac{z}{1}\Rightarrow \alpha :\beta :\gamma =1:1:1\Rightarrow \frac{\alpha }{\beta }+\frac{\beta }{\gamma }+\frac{\gamma }{\alpha }=3$