Abstract. The 3-Kronecker quiver is the quiver with two vertices, namely a sink and a source, and 3 arrows; its representations are given by 3 matrices of the same size, or, equivalently, by a 2-dimensional matrix pencil. It is the smallest (but a typical) wild matrix problem.
We will determine the elementary representtaions of the 3-Kronecker quiver. Let us recall that a regular representation of a quiver is said to be elementary provided it is non-zero and not a proper extension of regular representations. Of course any regular representation has a filtration whose factors are elementary. It turns out that the elementary 3-Kronecker modules are either tree modules or basic circle modules, thus determined by combinatorial invariants.