$1.$ Are trace $0$ matrices always of the form $AB-BA$?
$2.$ Is a trace $0$ matrix over the complex field always similar to a matrix with $0$ as a diagonal element?
$3.$ Is a trace $0$ matrices over any field always similar to a matrix with $0$ as a diagonal element?
$4.$ Is a trace $0$ matrix not invertible if it is upper triangular.?
I solved one problem in hoffman kunze saying : $W$ be the span of $n\times n$ matrices over the field $F$ and $W_0$ be the subspace spanned by the matrices $C$ where $C=AB-BA$. Then we proved there that $W_0$ is the exactly subspace of matrices which have trace $0$, so from this result can we say $1$ is true?