I just want to know how you decide whether to use the A-lambda method or by multiplying A by the vector (V1, V2) to get lambda(V1, V2) and then using substitution etc. when finding eigenvals/vecs. Thanks
I am not clear what the "A- lamba" method is or exactly what you are asking. I suspect you mean solving $\displaystyle det (A- \lambda I)= 0$. If so then, yes, that is the simplest way to find eigenvalues for reasonable sized matrices. For very large matrices there are various numerical methods that can be used on computers.