How to prove that tr(A) = tr(B) given that B is similar to A

Question

If A and B are similar, how does one prove that tr(A) = tr(B)

Answer 1

Hint: show that $\operatorname{tr}(CD) = \operatorname{tr}(DC)$

Answer 2

if $A$ and $B$ are similar, then $p(\lambda) = \det(A- \lambda I) = \det(B- \lambda I)$. therefore $$\operatorname{trace}(A) = \operatorname{trace}(B) = \text{coefficient of } \lambda^{n-1} \text{ in } p(\lambda) $$