  Given any commutative ring $R$, a commutator of two $n\times n$ matrices over
$R$ has trace $0$. In this paper, we study the converse: whether every $n
\times n$ trace $0$ matrix is a commutator. We show that if $R$ is a B\'{e}zout
domain with algebraically closed quotient field, then every $n\times n$ trace
$0$ matrix is a commutator. We also show that if $R$ is a regular ring with
large enough Krull dimension relative to $n$, then there exist a $n\times n$
trace $0$ matrix that is not a commutator. This improves on a result of Lissner
by increasing the size of the matrix allowed for a fixed $R$. We also give an
example of a Noetherian dimension $1$ commutative domain $R$ that admits a
$n\times n$ trace $0$ non-commutator for any $n\ge 2$.
