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?

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.

For more details, prefer this link:

link

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.

step-by-step explanation: