Let R be a commutative ring with identity, $${M_n(R)}$$ be the set of all $${n \times n}$$ matrices over R and $${M_n(R) ^{*} }$$ be the set of all non-zero matrices of $${M_n(R)}$$ where $${n \geq 2}$$ . For a matrix $${A \in M_n(R)}$$ , $${{\rm Tr} (A)}$$ is the trace of A. The trace graph of the matrix ring $${M_n(R)}$$ , denoted by $${\Gamma_t(M_n(R))}$$ , is the simple undirected graph denoted by $${\Gamma_t(M_n(R))}$$ with vertex set $$\{{A \in M_n(R) ^{*} : }$$ there exists $${B \in M_n(R) ^{*} }$$ such that $${{\rm Tr}(AB)=0}\}$$ and two distinct vertices A and B are adjacent if and only if $${{\rm Tr} (AB) = 0}$$ . First, we prove that $${\Gamma_t(M_n(R))}$$ is 2-connected and hence obtain Eulerian properties of $${\Gamma_t(M_n(R))}$$ . Also we obtain the domination number of $${\Gamma_t(M_n(R))}$$ of a commutative semisimple ring R and obtain the domination number for $${\Gamma_t(M_n(\mathbb Z_2^m))}$$ . Finally, it is proved that for a commutative ring R with identity, $${\Gamma_t(M_n(R))}$$ is non-planar and classified all finite commutative rings R with identity for which the trace graph has thickness 2.