If the eigenvalues for the characteristic equation are real and distinct, mathematically, nothing is really wrong. Thus, by our guess and the existence and uniqueness theorem, for an n-size square matrix, the solution set is determined by:

Then since the linear combination of two solutions is also a solution (which can be verified directly from the structure of the problem), we can form the general solution as such:

What's interesting is when the eigenvalues are not so simple.