Matrices and Networks - TCE Mathematics Applications (Tasmania)

Syllabus dot point

“Perform matrix operations and analyse graphs, paths and networks”

TCEMathematics ApplicationsUnit 38 min readUpdated 2 Jun 2026

Quick answer

You must add, subtract and multiply matrices, find the inverse of a 2x2 matrix, and use graphs and adjacency matrices to model networks, count paths and find shortest routes.

What this dot point is asking

This dot point joins two connected ideas: matrices as a way to store and combine numbers in a grid, and networks as diagrams of connected objects that can themselves be written as matrices.

Matrix operations

A matrix has order $m \times n$ (rows by columns). You can add or subtract two matrices only if they have the same order, by combining matching entries.

Matrix multiplication of $A$ ( $m \times n$ ) by $B$ ( $n \times p$ ) is defined only when the number of columns of $A$ equals the number of rows of $B$ . The result is $m \times p$ , and each entry is the dot product of a row of $A$ with a column of $B$ .

Multiplication is not commutative

In general $AB \neq BA$ . The order of multiplication matters, and sometimes only one order is even defined.

Matrix multiplication

Given two $2 \times 2$ matrices, we find their product $AB$ by computing the dot product of each row of $A$ with each column of $B$ .

Let $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$ .

Step 1: Understand the structure of the result

Because $A$ is $2 \times 2$ and $B$ is $2 \times 2$ , the product $AB$ is also $2 \times 2$ . Entry $(i, j)$ of $AB$ is the dot product of row $i$ of $A$ with column $j$ of $B$ : multiply matching entries and sum them.

Step 2: Compute each entry

Top-left entry (row 1 of $A$ , column 1 of $B$ ):

1 \times 5 + 2 \times 7 = 5 + 14 = 19

Top-right entry (row 1 of $A$ , column 2 of $B$ ):

1 \times 6 + 2 \times 8 = 6 + 16 = 22

Bottom-left entry (row 2 of $A$ , column 1 of $B$ ):

3 \times 5 + 4 \times 7 = 15 + 28 = 43

Bottom-right entry (row 2 of $A$ , column 2 of $B$ ):

3 \times 6 + 4 \times 8 = 18 + 32 = 50

Step 3: Write the result

AB = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}

Final answer: $AB = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}$ .

Determinant and inverse of a 2x2 matrix

For $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ , the determinant is $\det(A) = ad - bc$ . The inverse exists only when $\det(A) \neq 0$ :

A^{-1} = \frac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

2x2 inverse

Swap the leading diagonal, negate the other diagonal, then divide every entry by $\det(A) = ad - bc$ .

The inverse lets you solve a matrix equation $AX = B$ by computing $X = A^{-1}B$ .

Inverse of a 2x2 matrix and solving a system

We solve the simultaneous equations $2x + y = 5$ and $3x + 4y = 10$ by writing them as a matrix equation $AX = B$ and then multiplying both sides by $A^{-1}$ .

Step 1: Write the system in matrix form

Identify the coefficient matrix $A$ , the variable column $X$ , and the constant column $B$ . The left-hand side of each equation provides the row of $A$ , and the right-hand side gives the matching entry of $B$ :

A = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \end{pmatrix}, \quad B = \begin{pmatrix} 5 \\ 10 \end{pmatrix}

Step 2: Find the determinant of A

The determinant is the product of the leading diagonal minus the product of the other diagonal. A non-zero determinant confirms the inverse exists:

\det(A) = 2 \times 4 - 1 \times 3 = 8 - 3 = 5

Step 3: Compute the inverse

For a $2 \times 2$ matrix, swap the leading diagonal entries, negate the off-diagonal entries, and divide every entry by the determinant:

A^{-1} = \frac{1}{5}\begin{pmatrix} 4 & -1 \\ -3 & 2 \end{pmatrix}

Step 4: Solve by multiplying $A^{-1}B$

Because $AX = B$ implies $X = A^{-1}B$ , we multiply $A^{-1}$ by $B$ :

X = A^{-1}B = \frac{1}{5}\begin{pmatrix} 4 & -1 \\ -3 & 2 \end{pmatrix}\begin{pmatrix} 5 \\ 10 \end{pmatrix} = \frac{1}{5}\begin{pmatrix} 20 - 10 \\ -15 + 20 \end{pmatrix} = \frac{1}{5}\begin{pmatrix} 10 \\ 5 \end{pmatrix} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}

Substituting $x = 2$ and $y = 1$ back into either original equation confirms the solution.

Final answer: $x = 2$ and $y = 1$ .

Scalar multiplication scales every entry of a matrix by the same number, so $3A$ multiplies each entry of $A$ by $3$ . This is used when, for example, a transition or cost matrix is repeated a fixed number of times.

Graphs and networks

A graph (network) is a set of vertices (nodes) joined by edges. Key terms: the degree of a vertex is the number of edges meeting it; a path is a walk that does not repeat vertices; a connected graph has a route between every pair of vertices.

An adjacency matrix records how many edges directly join each pair of vertices. For an undirected graph this matrix is symmetric.

Adjacency matrix and 2-step paths

Counting walks

A network has vertices P, Q, R with edges P-Q, Q-R and P-R. Its adjacency matrix is:

M = \begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix}

The number of walks of length 2 between vertices is given by $M^2$ . Entry (P,P):

0\times0 + 1\times1 + 1\times1 = 2

There are two 2-step walks from P back to P (P-Q-P and P-R-P), matching the diagram.

Powers count walks

The entry in row $i$ , column $j$ of $M^k$ gives the number of walks of length exactly $k$ from vertex $i$ to vertex $j$ .

Shortest path

In a weighted network each edge carries a number (distance, cost or time). The shortest path between two vertices is the route with the smallest total weight. You find it by systematically labelling each vertex with the least cumulative distance from the start, then reading the smallest total at the destination and tracing back.

Check by listing routes

For small networks, list every sensible route and add the edge weights. The route with the lowest total is the shortest path. This is a fast safety check on more formal methods.

Confusing degree with number of edges

The degree of a vertex counts edge-ends meeting it, so a loop (an edge from a vertex to itself) adds 2 to the degree, not 1. Also, the sum of all vertex degrees equals twice the number of edges, never the number of edges. Use this handshake relationship to check your edge count.

In the exam, watch the order of matrix multiplication, confirm the determinant is non-zero before inverting, and use matrix powers to count walks rather than tracing every path by hand.

Exam-style questions

Questions in the style of TASC exam questions on this dot point, each with a worked answer. They are written by ExamExplained unless tagged "Past paper"; the year shows the paper a question is modelled on.

Practice1 mark

A graph shows the network of walking tracks in a national park (Ranger Station A and camps B to G), with track lengths marked. The ranger wishes to walk every track once, without repeating any track, starting and finishing at the Ranger Station. What mathematical name is given to this type of route?

Show worked answer →

(1 mark) A route that travels every edge (track) of a network exactly once and returns to its starting vertex is an Eulerian circuit (an Eulerian trail is accepted when start and finish differ). Because the ranger starts and finishes at the same vertex (the Ranger Station) and uses every track once, the correct name is an Eulerian circuit.

Practise this

Sources & how we know this

General Mathematics Level 3 (MTG315123) course document (2023 onwards) — TASC

What this dot point is asking

Matrix operations

Step 1: Understand the structure of the result

Step 2: Compute each entry

Step 3: Write the result

Determinant and inverse of a 2x2 matrix

Step 1: Write the system in matrix form

Step 2: Find the determinant of A

Step 3: Compute the inverse

Step 4: Solve by multiplying A−1BA^{-1}BA−1B

Graphs and networks

Counting walks

Shortest path

Exam-style questions

Practise this

Sources & how we know this

Step 4: Solve by multiplying $A^{-1}B$