  Over a field of characteristic 0, the algebra of invariants of several
$n\times n$ matrices under simultaneous conjugation by $GL_n$ is generated by
traces of products of generic matrices. In this paper we have found, in terms
of representation theory of $GL_2$, a minimal set of generators of the algebra
of invariants of two $4\times 4$ matrices. The proof is purely combinatorial
and involves computer calculations with standard functions of Maple.
