Why does $\mathrm{Rank}(A{A^*} - {A^*}A) \ne 1$? Given $A \in M_n$, why does $\mathrm{Rank}(A{A^*} - {A^*}A) \neq 1$?

Hint. 
Observe that
$$
\mathrm{Trace}(AA^*)=\mathrm{Trace}(A^*A),
$$
and hence
$$
\mathrm{Trace}(AA^*-A^*A)=0.
$$
Next show that, if $B$ is diagonalizable and $\mathrm{Rank}(B)=1$, then $\mathrm{Trace}(B)\ne 0$.
Finally, observe that $AA^*-A^*A$ is hermitian and hence diagonalizable.
