Application of Matrices (Vertex-Edge Graphs and
Communication Matrices)
Warm-Up
Use Elimination to solve the given system of equations:
5x + 4y = -2
2x + 3y = -5
Essential Question
How do you use matrices to represent and solve realistic
situations using vertex-edge graphs?
Vertex-Edge Graph: Collection of points and line segments
conecting some (possibly empty) subset of the points.
Edge: It is a line segment connecting the vertices of a
graph.
Vertex: It is a point that is either the endpoint of an edge
or not part of an edge.
Digraph (Directed Network): Graph with vertices and
arrows that represent the direction in which the graph flows.
Adjacency (Communication) Matrix: Models a
communication network. In such a matrix, a “1” in any row
indicates that direct communication from vertex to another.
A “0” indicates that direct communication is not possible.
Examples:
1. Use a Vertex-Edge graph to represent the situation, and
then write a matrix that represents the vertex-edge graph.
An airline serves four cities: Bedford, Columbia, Dunwich,
and Exton. There are flights between Bedford and Columbia,
Bedford and Dunwich, and Columbia and Exton.
Step 1: Represent the cities using points.
Step 2: Draw the edges for the graph.
Step 3: Write the matrix that represents the vertex-edge
graph.
You Try:
2. An airline serves five cities: Lowell, Montour, Newman,
Peoria, and Orlando. There are flights between Lowell and
Montour, Lowell and Orlando, Montour and Orlando, Newman
and Orlando, and Newman and Peoria. Draw a vertex-edge
graph and write the matrix that represents this situation.
Directed Network
J
K
L
From:
Adjacency Matrix
To:
J
K
L
J 0
1
0
K 0
0
2
L 1
2
1
If A is the adjacency matrix, then A x A = A2 is the
number of two-stage paths from one vertex to another
vertex by means of an intermediate vertex.
Example Problems.
1. a. Write an adjacency matrix.
b. Find A2.
c. Interpret A3,2.
J
K
L