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Subjects · Leaving Cert Applied Maths

Leaving Cert Applied Maths: Matrices & adjacency

How often Matrices & adjacency comes up on the Applied Maths papers, every year it was asked, and questions to try.

HL Asked on 3 of the last 3 Higher Level papers, most recently in 2025. banker

OL Asked on 3 of the last 3 Ordinary Level papers, most recently in 2025. banker

Matrices & adjacency, Higher Level(7 marks)

Quick ones on Matrices & adjacency.

(a)The adjacency matrix of an undirected graph is always what?
  1. The identity matrix
  2. Symmetric
  3. Upper triangular
(b)What does the entry in row i, column j of M² count?
  1. Walks of length 1 or 2 from i to j
  2. Edges joining i directly to j
  3. Walks of length 2 from i to j
(c)Undirected graph: edges A–B, B–C, C–A, C–D. Row C of its adjacency matrix (order A, B, C, D)?
  1. 1 1 0 1
  2. 0 0 1 1
  3. 1 1 1 1
Show the answers

(a) Symmetric

(b) Walks of length 2 from i to j

(c) 1 1 0 1

Your turn: Higher Level questions on Matrices & adjacency.

Higher Level

Asked on 3 of the last 3 Higher Level papers, most recently in 2025. banker

Every paper, year by year

YearWhere it came up
2025Q1
2024Q1
2023Q1

Links open the State Examinations Commission’s paper for that year.

More Matrices & adjacency questions

Matrices & adjacency, 2 marks

What does the entry in row i, column j of M³ count?

  1. Paths of length 3 with no repeated vertex
  2. Triangles that contain both i and j
  3. Walks of length 3 from i to j
Show the answer

Walks of length 3 from i to j

Each power of M adds one step: Mᵏ counts walks of length k. Walks may repeat vertices and edges, so M³ counts more than just paths or triangles.

Matrices & adjacency, 2 marks

Undirected graph, no loops: what does the sum of one row of its adjacency matrix give?

  1. Twice the number of edges
  2. The degree of that vertex
  3. The number of vertices
Show the answer

The degree of that vertex

Row i lists the edges from vertex i to each other vertex, so its total is the number of edges at i: its degree. Summing the whole matrix gives twice the number of edges.

Matrices & adjacency, 3 marks

Simple undirected graph: what does the diagonal entry (i, i) of M² give?

  1. The degree of vertex i
  2. Zero
  3. The number of edges in the graph
Show the answer

The degree of vertex i

(i, i) of M² counts walks of length 2 from i back to i: out along an edge and straight back. There is one such walk for each edge at i, so it equals the degree.

Ordinary Level

Asked on 3 of the last 3 Ordinary Level papers, most recently in 2025. banker

Every paper, year by year

YearWhere it came up
2025Q9
2024Q5
2023Q1

Links open the State Examinations Commission’s paper for that year.

More Matrices & adjacency questions

Matrices & adjacency, 2 marks

What does an entry of M² count, where M is an adjacency matrix?

  1. Edges joining the two vertices directly
  2. Walks of length 1 between the two vertices
  3. Walks of length 2 between the two vertices
Show the answer

Walks of length 2 between the two vertices

M counts one-step walks. M² = M × M counts two-step walks, so an entry of M² tells you how many ways to get from one vertex to another in exactly 2 edges.

Matrices & adjacency, 2 marks

In the adjacency matrix of a simple undirected graph, the sum of row A equals…

  1. The number of cycles through A
  2. The degree of A
  3. The number of vertices
Show the answer

The degree of A

Each 1 in row A is an edge at A, so adding the row counts the edges meeting at A, which is its degree.

Matrices & adjacency, 3 marks

An undirected graph has edges AB, AC and BC. Row A of its adjacency matrix (order A, B, C)?

  1. 0 1 1
  2. 1 1 1
  3. 0 1 0
Show the answer

0 1 1

A has no loop, so the A column entry is 0. A is joined once to B and once to C, giving 0 1 1.

Other Applied Maths topics

All of Leaving Cert Applied Maths