Question

2. Prove that a 3x3 matrix must have at least one real eigenvalue. Is it important, as part of the proof, to find this eigenvalue? Why or why not?

Answer 1

A 3x3 matrix has a characteristic polynomial of degree 3. If all the elements of the matrix are real, then the polynomial has up to 3 distinct complex roots. If one of these roots is complex (in particular, has a non-zero imaginary part), then a second root would be that first root's complex conjugate. Then the remaining root has to be real.