I am confused on something that I have not been able to find a good answer for: How do I determine in a vector has real, real repeating, or complex eiganvalues? When I have asked, and read my...

I am confused on something that I have not been able to find a good answer for:

How do I determine in a vector has real, real repeating, or complex eiganvalues? When I have asked, and read my textbook, I dont get a clear answer.

Can you please explain using the vectors in the image below?