Characteristic polynomial roots of $ 3\times 3$ orthogonal matrix.

Question

How would you approach this problem?

Let $A$ be an orthogonal $3$ by $3$ matrix. That is, $A^TA = AA^T=I_3$. Prove that the characteristic polynomial $\textit{p}_A$ has a real root.

I am not familiar with how to prove a third degree polynomial has a real root. I started the problem by noticing that $\det(A-tI) = \det(A-tAA^T)=\det(A)\det(I-tA^T)$, but this isn't getting me anywhere.

Answer 1

As you mentioned the characteristic polynomial of your matrix is a third degree polynomial and every third degree polynomial has a real root.

You do not have to prove that every third degree polynomial has a real root to solve your problem because it is a well-known fact by just looking at the end behavior of the polynomial.