 Solution

On the line spanned by $n$, $A$ multiplies by $L=\alpha +\beta =a+b+c$. On $n^{\perp }$, it is the composition of a rotation with a dilation by $r=\sqrt{{\alpha }^2+{\gamma }^2}$. Consequently every area in that plane is multiplied by $r^2={\alpha }^2+{\gamma }^2$.