Year 11 General Matrices And Matrix Arithmetic

Communications And Connections

20 practice questions 2 video lessons Theory + worked examples

Theory

A communication matrix records direct connections (1) or no connection (0) between pairs of people or places. Two-way networks give symmetric matrices with zero diagonals. A row sum counts one person's direct connections; total edges = (sum of all entries) ÷ 2. The power $M^{2}$ counts two-step paths.

A communication matrix (also called a connection or adjacency matrix) records which pairs of people, places, or things are directly connected. Entries are usually $1$ (a connection exists) or $0$ (no connection).

Number the people or places $1$ to $n$ in some chosen order. Then $M_{i j} = 1$ if person $i$ is connected to person $j$ , and $0$ otherwise.

The matrix has these key properties:

Two-way (undirected) networks have symmetric matrices: $M_{i j} = M_{j i}$ .
One-way (directed) networks (like one-way streets) need not be symmetric.
The main diagonal is usually all $0$ s — no self-connections.
Row sum for row $i$ = number of direct connections person $i$ has.
Total edges in a two-way network = (sum of all entries) $\div$ 2.

Crucially, the power $M^{2}$ counts two-step paths: $(M^{2})_{i j}$ is the number of paths of length 2 from $i$ to $j$ , going through one intermediate person.

The first diagram pairs a small 4-person network with its adjacency matrix. The second shows why $M^{2}$ counts two-step paths by tracing the two routes from Alice to Dan.

Network of A, B, C, D and its symmetric

4 \times 4

adjacency matrix. Diagonal = 0.

Two paths from Alice to Dan via Ben and via Carla —

(M^{2})_{14} = 2

The key formulas are simple counting rules.

Building the matrix

$M_{i j} = {\begin{cases} 1 & if i is directly connected to j \\ 0 & otherwise \end{cases}$

M_{i j} = 1 if connected, 0 otherwise

Counting connections

Quantity	How to compute
Direct connections of person $i$	row sum of row $i$
Total edges (two-way network)	(sum of all entries) $\div$ 2
Two-step paths from $i$ to $j$	$(M^{2})_{i j}$
Three-step paths from $i$ to $j$	$(M^{3})_{i j}$
Isolated person	row of all zeros

The power trick. If M records direct (one-step) connections, then M2 counts two-step paths, M3 counts three-step paths, and so on. A diagonal entry of M2, say (M2)ii, counts how many "round-trips" leave i and return — equivalently, how many direct connections i has.

Building a communication matrix

Number the people or places in your chosen order (alphabetical is common).
Use that same order for both rows and columns.
Set diagonal entries to 0 — no self-connections.
For each pair, write 1 if directly connected, 0 if not. For a two-way network, this means $M_{i j} = M_{j i}$ .

Counting individual and total connections

Person's connections: add the entries in that person's row.
Total edges: add every entry in the matrix, then divide by 2.
Isolated person: look for a row (and column) of all zeros.

Counting two-step paths

Compute $M^{2} = M \times M$ using the row-by-column rule.
The entry $(M^{2})_{i j}$ is the number of two-step paths from $i$ to $j$ .
To check by inspection: list which people $i$ connects to directly, then count how many of them connect directly to $j$ .

EXAMPLE 1 — READ DIRECT CONNECTIONS

For the matrix

M = [\begin{matrix} 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \end{matrix}]

(rows/columns ordered A, B, C, D), how many people does Ben communicate with directly?

SOLUTION

Ben is row 2. Sum the entries of row 2.

Row 2 sum	$=$	$1 + 0 + 0 + 1$
	$=$	$2$

Answer: Ben communicates with $2$ people directly — Alice and Dan.

Ben's connections = 2

EXAMPLE 2 — TOTAL EDGES

For the same matrix

M

, how many distinct two-way communication links does the network have?

SOLUTION

Sum all entries in the matrix, then halve the total (each edge is counted twice in a symmetric matrix).

Sum of all entries	$=$	$8$
Total edges	$=$	$\frac{8}{2} = 4$

Answer: the network has $4$ distinct two-way links — A–B, A–C, B–D, and C–D.

edges = \frac{8}{2} = 4

EXAMPLE 3 — TWO-STEP PATHS VIA M²

For the same matrix

M

, find

(M^{2})_{14}

, the number of two-step paths from Alice to Dan.

SOLUTION

A two-step path Alice → ? → Dan goes through an intermediate person. Alice is directly connected to Ben (B) and Carla (C). Are they connected to Dan?

From the matrix: $M_{24} = 1$ (Ben → Dan ✓), and $M_{34} = 1$ (Carla → Dan ✓). So there are two paths:

Path 1	:	$A \to B \to D$
Path 2	:	$A \to C \to D$

Answer: $(M^{2})_{14} = 2$ . There are two two-step paths from Alice to Dan.

(M^{2})_{14} = 2

EXAMPLE 4 — IDENTIFY ISOLATED PERSON

For

N = [\begin{matrix} 0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}]

, identify which member is isolated (has no direct connections).

SOLUTION

An isolated person has a row of all zeros (and, since the matrix is symmetric, a column of all zeros too).

Row 1 sum	$=$	$0 + 1 + 1 + 0 = 2$
Row 2 sum	$=$	$1 + 0 + 1 + 0 = 2$
Row 3 sum	$=$	$1 + 1 + 0 + 0 = 2$
Row 4 sum	$=$	$0 + 0 + 0 + 0 = 0$

Answer: member 4 has row sum $0$ , so member 4 is isolated.

Isolated = member 4

Common pitfalls

The diagonal of $M$ is 0, but the diagonal of $M^{2}$ is usually not.

M_{i i} = 0

because a person doesn't communicate with themselves. But

(M^{2})_{i i}

counts round-trip two-step paths, which equals the number of direct connections that person has.

One-way vs two-way networks. For two-way (undirected) networks,

M

is symmetric. For one-way (directed) networks, it isn't. Read the question carefully — "talks to", "links to", "follows" can all be one-way or two-way depending on context.

Total edges = (sum of all entries) ÷ 2. Don't forget the ÷ 2. Each edge appears twice in a symmetric matrix — once at

(i, j)

and again at

(j, i)

— so dividing by 2 gives the correct count.

"Direct" vs "indirect" connections. Direct = one step (read off

M

). Two-step indirect via one intermediate = entries of

M^{2}

. Don't confuse the two.

A row sum of 0 means isolated. If person

i

has no entries equal to 1 in their row, they have no direct connections at all. (For a symmetric matrix, their column will also be all zeros.)

Frequently asked questions

What is a communication matrix?

A communication matrix (also called a connection or adjacency matrix) records which pairs of people, places, or things are directly connected. Each entry is usually 1 (a direct connection exists) or 0 (no direct connection).

What does an entry of 1 versus 0 mean?

Entry (i, j) equals 1 if person i has a direct connection to person j, and 0 if not. The main diagonal is usually 0 because people are not considered to communicate with themselves.

Why is the matrix usually symmetric?

For two-way (undirected) networks like face-to-face conversation or two-way roads, if A is connected to B then B is automatically connected to A. So entry (i, j) equals entry (j, i), making the matrix symmetric. One-way (directed) networks are NOT symmetric.

What does the row sum tell me?

The sum of row i counts how many direct connections person i has. A row of all zeros means that person is isolated — no direct connections to anyone.

How do I count the total number of edges in a two-way network?

Add up every entry in the matrix and divide by 2. The division by 2 is because each edge is counted twice (once at entry (i, j) and once at entry (j, i)).

What does M squared tell me?

If M is the direct (one-step) communication matrix, then M squared counts two-step paths: the entry (i, j) in M squared is the number of paths of length 2 from person i to person j, going through one intermediate person. M cubed counts three-step paths, and so on.

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Communications And Connections

📖 Theory

Building the matrix

Counting connections

Building a communication matrix

Counting individual and total connections

Counting two-step paths

Common pitfalls

Frequently asked questions

🎬 Video Lessons

✏️ Practice Questions

Theory

Video Lessons

Practice Questions