If A is a an n x n matrix then det(A - sI) is an nth degree polynomial. The zeros of this polynomial are the eigen values for the system. So, to find the eigen values for a system means solving a polynomial equation. This is generally not too computationally difficult if n is kept small, but nothing says we have to keep n small.
For this prompt, first discuss all types of eigen values possible. That is, suppose s1, s2, s3, ..., sn is the complete list of solutions to the characteristic polynomial det(A - sI) = 0. (Note: the list includes all zeros of the polynomial, counting their repetition, if necessary.) Then answer: Is it possible for a system of linear first order differential equation to not have associated eigenvalues? Why or why not?