  In this paper natural necessary and sufficient conditions for quantifier
elimination of matrix rings $M_n(K)$ in the language of rings expanded by two
unary functions, naming the trace and transposition, are identified. This is
used together with invariant theory to prove quantifier elimination when $K$ is
an intersection of real closed fields. On the other hand, it is shown that
finding a natural \textit{definable} expansion with quantifier elimination of
the theory of $M_n(\C)$ is closely related to the infamous simultaneous
conjugacy problem in matrix theory. Finally, for various natural structures
describing dimension-free matrices it is shown that no such elimination results
can hold by establishing undecidability results.
