Let $A$ be an $n \times n$ matrix over $\mathbb{C}$. What does it mean to say that $\lambda$ is an eigenvalue of $A$ ? Show that $A$ has at least one eigenvalue. For each of the following statements, provide a proof or a counterexample as appropriate.

(i) If $A$ is Hermitian, all eigenvalues of $A$ are real.

(ii) If all eigenvalues of $A$ are real, $A$ is Hermitian.

(iii) If all entries of $A$ are real and positive, all eigenvalues of $A$ have positive real part.

(iv) If $A$ and $B$ have the same trace and determinant then they have the same eigenvalues.