  Over a field of characteristic 0, the algebra of invariants of several
$n\times n$ matrices under simultameous conjugation by $GL_n$ is generated by
traces of products of generic matrices. Teranishi, 1986, found a minimal system
of eleven generators of the algebra of invariants of two $3\times 3$ matrices.
Nakamoto, 2002, obtained an explicit, but very complicated, defining relation
for a similar system of generators over $\mathbb Z$. In this paper we have
found another natural set of eleven generators of this algebra of invariants
over a field of characteristic 0 and have given the defining relation with
respect to this set. Our defining relation is much simpler than that of
Nakamoto. The proof is based on easy computer calculations with standard
functions of Maple but the explicit form of the relation has been found with
methods of representation theory of general linear groups.
