  The trace algebra C(n,d) over a field of characteristic 0 is generated by all
traces of products of d generic nxn matrices, n,d>1. Minimal sets of generators
of C(n,d) are known for n=2 and n=3 for any d as well as for n=4 and n=5 and
d=2. The defining relations between the generators are found for n=2 and any d
and for n=3, d=2 only. Starting with the generating set of C(3,d) given by
Abeasis and Pittaluga in 1989, we have shown that the minimal degree of the set
of defining relations of C(3,d) is equal to 7 for any d>2. We have determined
all relations of minimal degree. For d=3 we have also found the defining
relations of degree 8. The proofs are based on methods of representation theory
of the general linear group and easy computer calculations with standard
functions of Maple.
