Year 12 Discrete Mathematics: Communication and dominance matrices

Communication and dominance matrices are discrete models for relationships inside a network or a group. Instead of tracking population or movement, they track who can reach whom or who outranks whom. These questions reward careful interpretation because the entries are usually about relations, not quantities. A $1$ may mean a direct connection, a message path, or a dominance relation, depending on the context. That means the matrix arithmetic has to be read through the meaning of the relation, not just performed mechanically.

In relational matrices, always state what a

1

means before you multiply or interpret a power of the matrix.

Subtopic 1: Communication matrices

A communication matrix typically records which nodes in a network are directly linked. Powers of the matrix or related constructions can then show indirect connections through one or more steps. The point is to discover reachability, not simply to produce a product. Once you understand that, the entries in the result become meaningful statements about paths in the network.

Worked example 1

Problem: Three people are represented by the relation matrix

A = [[0, 1, 0], [0, 0, 1], [0, 0, 0]]

, where a

1

means "can send directly to." Explain what the entry in row 1, column 3 of

A^{2}

represents.

The matrix $A$ records one-step communication.
The square $A^{2}$ records two-step communication paths.
Row 1, column 3 asks whether person 1 can reach person 3 in two steps.
Because person 1 can send to person 2 and person 2 can send to person 3, such a path exists.

Answer: the entry represents whether person 1 can communicate with person 3 through exactly two direct links, and here the answer is yes.

Subtopic 2: Dominance matrices

A dominance matrix records which participant defeats or dominates another. The interpretation is again relational rather than numerical. Rows and columns represent competitors or options, and a $1$ usually means the row entity dominates the column entity. The matrix structure can reveal patterns of superiority, indirect chains of dominance, or incomparability.

Worked example 2

Problem: Suppose

D = [[0, 1, 1], [0, 0, 1], [0, 0, 0]]

, where a

1

means "dominates." What does row 1 tell you?

Row 1 is $[0, 1, 1]$ .
The first zero means entity 1 does not dominate itself.
The two ones mean entity 1 dominates entity 2 and entity 3.

Answer: entity 1 dominates both entity 2 and entity 3 directly.

Relational meaning of matrix powers

In communication questions, higher powers often mean communication through multiple steps. In dominance questions, they may indicate chains of influence or multi-stage superiority. The precise interpretation depends on the model, which is why a verbal explanation is so important. A matrix product on its own is not the goal. The goal is to determine what kind of relationship has been created or detected after one or more steps in the system.

Worked example 3

Problem: In the communication matrix

A = [[0, 1, 0], [0, 0, 1], [0, 0, 0]]

, why does person 1 communicate indirectly with person 3 even though

a_{13} = 0

The entry $a_{13} = 0$ means there is no direct one-step link from 1 to 3.
However, communication can still happen through an intermediate person.
Here person 1 communicates with person 2, and person 2 communicates with person 3.
That creates an indirect two-step path, which appears in the communication analysis based on matrix powers.

Answer: person 1 reaches person 3 through person 2, so the relationship exists indirectly even though it is not direct.

Common traps

Forgetting to define what a $1$ means in the relation.
Treating the matrix as if it were about quantities rather than relationships.
Interpreting a power of the matrix without connecting it to multi-step paths.
Reversing row and column meaning in a network context.

Revision focus

When revising, attach a small verbal sentence to each important entry. For example, say "row 2, column 3 means entity 2 dominates entity 3" or "node 2 can send directly to node 3." This habit keeps the relational meaning visible and stops the matrix from becoming a meaningless pattern of ones and zeros.

It also helps to draw a quick network diagram beside the matrix. Seeing the same relation as both a graph and a matrix makes indirect paths and dominance chains much easier to interpret. In discrete mathematics, multiple representations often reinforce each other strongly.