How many eigenvalues does a 2 by 2 matrix have?

There can be at most n. If the field of scalars is algebraically closed, as in the complex numbers, then all the irreducible factors will be linear, so there will be at least one eigenvalue. It is easy to construct a real 2 by 2 matrix whose minimal polynomial is x^2 + 1. It obviously has no eigenvalues.
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