Graph theory has existed for many years not only as an area mathematical study but also as an intuitive and illustrative tool. The use graphs in wiring diagrams is a straightforward representation the physical elements an electrical circuit; a street map is also a graph with the streets as edges, intersections streets as vertices, and street names as labels the edges. The graphs resemble the physical object that they represent in these cases, and so the application (and sometimes the genesis) the graph-theoretic ideas is immediate. A flow diagram a computer program and a road map with one way streets are examples graphs which contain the concept direction or flow to the edges; these are called directed graphs. There are applications graphs and directed graphs in almost all areas the physical sciences and mathematics, many them known for fifty years or more, but very few these ideas have percolated down to the undergraduate student. The purpose this article is to apply graph-theoretic ideas to some the fundamental topics in linear algebra. While there are many such applications, we shall focus on only two the most elementary ones, i.e., matrix multiplication and the theory determinants. The presentation is usable as a supplement to the usual classroom lectures; in fact, this paper grew out notes used in a linear algebra course for sophomores. The main tool to be used is the directed graph. Intuitively this can be thought as a set points (or vertices) with arrows (or arcs) joining some the points. A label may be put on an arc. More formally, a digraph consists a set vertices V and a subset ordered pairs vertices called the arcs. A labelling the digraph is a function from the arcs to the real numbers. A labelled digraph is usually visualized by considering the vertices as points with arcs as arrows going from vertex i to vertexj whenever (i, j) belongs to the sets arcs. The ith vertex the arc (i, j) is called its initial vertex, while the jth vertex is called its terminal vertex. The arc is then given a label which is the image that arc under the labelling function. When the initial and terminal vertices are identical, the arc is called a loop. We shall sometimes say that an arc goes out of its initial vertex and that it goes its terminal vertex. The number arcs that go out a vertex is called its outdegree, and the number arcs that go into that vertex is called its