Undergrad Normal Mode calculation steps

[vgarg]

Can someone please explain me the steps of calculation of X1:X2 after putting in the lower value of W^2 in equation 9.9 in "Riley, Hobson, Bence - Mathematical Methods for Physics and Engineering 2006 - pg 319"? I have attached the page as a PDF file.
Thank you.

 

[Office_Shredder]

A and B are matrices you know, and $\,omega$ is a number that you know, so you can plug them all in and you get like like $Cx=0$ for some matrix $C$ that you know. You might at first guess that gives you two equations and two unknown ($x$ is a vector of 2 dimensions, the 0 on the right side is also a vector of 2 dimensions so you get two equations) but if $x$ is a solution so is $\alpha x$ for any scalar $\alpha$, which means the two equations are not independent, and there are infinite solutions. So all you can do is express one variable in terms of the other which is the same as computing their ratio

 

[vgarg]

A and B are matrices you know, and $\,omega$ is a number that you know, so you can plug them all in and you get like like $Cx=0$ for some matrix $C$ that you know. You might at first guess that gives you two equations and two unknown ($x$ is a vector of 2 dimensions, the 0 on the right side is also a vector of 2 dimensions so you get two equations) but if $x$ is a solution so is $\alpha x$ for any scalar $\alpha$, which means the two equations are not independent, and there are infinite solutions. So all you can do is express one variable in terms of the other which is the same as computing their ratio

Thank you very much.