Consider a variable matrix $V=\left[\begin{array}{ccc}x& y& z\\ y& z& x\\ z& x& y\end{array}\right]$ such that $VA=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$ _______(1), 

where $A=\left[\begin{array}{c}\log a\\ \log b\\ \log c\end{array}\right]$ and a, b, c are distinct positive reals in G.P.. If solutions (x, y, z) of matrix equation (1) lie on line $\frac{x}{\alpha }=\frac{y}{\beta }=\frac{z}{\gamma },$ then $\frac{\alpha }{\beta }+\frac{\beta }{\gamma }+\frac{\gamma }{\alpha }$ is $(x,y,z\ne 0)$

$\therefore VA=0$ (Null matrix)

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

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

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

which is homogenesis system of linear equations in x, y and z.

$\because x,y,z\ne 0$

$\Rightarrow$system possesses non trivial solutions.

$\Rightarrow D=\left|\begin{array}{ccc}\log a& \log b& \log c\\ \log b& \log c& \log a\\ \log c& \log a& \log b\end{array}\right|=0\Rightarrow \log a+\log b+\log c=0$ OR $\log a+\log b+\log c$

$\because a,b,c,$ are distinct

$\Rightarrow \log a+\log b+\log c=0$

$\Rightarrow$a b c = 1$\Rightarrow b^3=1(\because b^2=ac)\Rightarrow b=1\Rightarrow ac=1\Rightarrow c=\frac{1}{a}$

$\Rightarrow$System become

(log a) x – (log a ) z = 0, - (log a) y + (log a) z = 0, -(log a) x – (log a) y = 0

$\Rightarrow x-z=0, y-2=0, x-y=0$

$\Rightarrow x=y=2\Rightarrow \alpha :\beta :\gamma =1:1:1$

$\frac{\alpha }{\beta }+\frac{\beta }{\gamma }+\frac{\gamma }{\alpha }=3$