Trace of square of a rank-$1$ Hermitian matrix ${\bf A} = {\bf a}{\bf a}^H$

Question

Given matrix ${\bf A} = {\bf a}{\bf a}^H$, where ${\bf a}$ is a complex column vector. Are the following correct?

1. ${\rm Tr}({\bf A}) = {\rm Tr} \left( {\bf a}{\bf a}^H \right) = {\rm Tr}({\bf a}^H{\bf a}) = \|{\bf a}\|^2$.

2. ${\rm Tr}\left({\bf A}^2\right) = {\rm Tr}([{\bf a}{\bf a}^H][{\bf a}{\bf a}^H]) = {\rm Tr}({\bf a}[{\bf a}^H{\bf a}]{\bf a}^H) = {\rm Tr}({\bf a}\|{\bf a}\|^2{\bf a}^H) = \|{\bf a}\|^2{\rm Tr}({\bf a}{\bf a}^H) = \|{\bf a}\|^2{\rm Tr}({\bf a}^H{\bf a}) = \|{\bf a}\|^4$.

Please help me confirm if I am right or wrong. Your inputs are highly appreciated.