Matrix trace, are those statements right?

Question

Is the following statements true?

$trace(AB)=trace(BA)$

$trace(AB)=trace((AB)^T))$

$trace(A)=trace(A^{-1})$

I believe that the third isn't (Plus, I hope that someone could mention some important statements regarding trace in addition to what I wrote)

Answer 1

Trace of a matrix is sum of its eigenvalues (with multiplicity). And eigenvalues of $A^{-1}$ are of the form $\displaystyle\frac1{\lambda}$, where $\lambda$ is an eigenvalue of $A$.

So $\operatorname{Tr}(A)=\operatorname{Tr}(A^{-1})$ implies
$\sum\lambda=\sum\displaystyle\frac1\lambda$, which is not always true.