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.

Answer 2

In this  question, we have to prove that a 3x3 matrix must have at least one real eigenvalue.

Given:

A  3x3 matrix .

Prove :

A 3x3 matrix must have at least one real eigenvalue.

A 3x3 matrix has a characteristic polynomial of degree 3.

If all the elements of the matrix are real, then the polynomial has 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 the first root's complex conjugate. Then, the remaining root has to be real.

Hence proved.

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