Finding natural frequencies with Eigenvalues help!

Here is the assignment and some of the work I've done on it. I'm really not sure if I'm going in the right direction. The examples done in class were nothing like this.

Does my work on problem 1a look correct? If so, what is my next step? Do I solve for omega? Am I solving for it right? It doesn't seem right because nothing I've done looks like finding an Eigenvalue or Eigenvector.

Please help me by sending me in the right direction (I will also need to program this into MATLAB but if I can do it by hand it shouldn't be a problem putting in MATLAB)

If I am doing this completely wrong please refer me to a place I can get help or give me a few steps to start.

Thanks in advance!