The idea, step by step
Choose a convention before entering data
For adjacency matrices here, row means starting vertex and column means destination. An entry of 1 means one directed connection; 0 means none. The diagonal records loops. Another source may use the transpose, so the convention must be stated rather than guessed.
A directed link A to B affects row A, column B only. An undirected link appears in both directions, giving a symmetric matrix. Symmetry does not imply every possible link exists.
Labels determine the model
Keep the same vertex order for rows and columns before taking powers. Reordering vertices is valid only if the rows and columns are reordered together. Otherwise a multiplication may join the wrong intermediate labels.
Weights might count bus services, measure travel times or describe probabilities. These are different models. Powers of a matrix whose entries count connections count walks; they do not automatically find the shortest travel time.
| From / to | A | B | C |
|---|---|---|---|
| A | 0 | 1 | 1 |
| B | 0 | 0 | 1 |
| C | 1 | 0 | 0 |
A common trap
Reversing the arrow when reading row A, column B.
Instead: With row=start, that entry records A→B.
See the thinking
Read a directed network
List the arrows in the adjacency table above.
- Row A has 1 in columns B,C.
- Row B has 1 only in column C.
- Row C has 1 only in column A.
An undirected example
Rows in A,B,C order are (0,1,0),(1,0,1),(0,1,0). Describe the network.
- The matrix is symmetric.
- Its off-diagonal pairs give A–B and B–C, with no A–C link.
Check your understanding
Where is C→B recorded in row-start convention?
Show answer
Row C, column B.
The row gives the starting point.
Your turn
6 questions · 20 marks · Work at your own pace.
Read the questions online
- Construct the adjacency matrix, ordered A,B,C, for A→B, B→A, B→C and C→C.3 marks
- Rows P,Q,R are (0,2,0),(0,0,1),(1,0,0). Entries count direct bus services. Interpret the 2.3 marks
- Rows A,B,C are (0,1,1),(1,0,0),(1,0,0). Can this represent an undirected network? List its links.3 marks
- Explain why zero diagonal entries do not rule out cycles.3 marks
- Use the diagram and vertex order A,B,C.5 marks
- Someone swaps the first two rows of an A,B,C adjacency matrix but not its columns, then calls both orders B,A,C. Explain the error.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Construct the adjacency matrix, ordered A,B,C, for A→B, B→A, B→C and C→C.3 marks
Place a 1 for every stated arrow.
C→C belongs on the diagonal.
Rows (0,1,0),(1,0,1),(0,0,1). - Rows P,Q,R are (0,2,0),(0,0,1),(1,0,0). Entries count direct bus services. Interpret the 2.3 marks
Use row P and column Q.
The weight is a count, not a probability or a duration.
Two direct services from P to Q. - Rows A,B,C are (0,1,1),(1,0,0),(1,0,0). Can this represent an undirected network? List its links.3 marks
The matrix is symmetric; each link appears in both directions.
Yes: A–B and A–C. - Explain why zero diagonal entries do not rule out cycles.3 marks
A→B→A is a two-step return without either vertex having a loop.
They rule out one-step loops only. Longer walks can return to the start. - Use the diagram and vertex order A,B,C.5 marks
a) Read all arrows leaving each vertex.
b) Sum row A and column C respectively.
a) Rows (0,1,1),(0,0,1),(1,0,0).; b) Two leave A; two arrive at C. - Someone swaps the first two rows of an A,B,C adjacency matrix but not its columns, then calls both orders B,A,C. Explain the error.3 marks
The old column A is wrongly being treated as column B.
Its row and column labels no longer agree. Both must be reordered consistently.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 2 General Mathematics matrix-models option. This is the offered fifth-topic option, alongside the four compulsory topics; a school-designed Open Topic is an alternative, not prescribed content in this chapter.
Official curriculum source ↗