QCE Specialist Mathematics - Unit 3 - Further matrices

Applications of Matrices | QCE Specialist Mathematics

Learn QCE Specialist dominance matrices, Leslie matrices, matrix powers and practical modelling assumptions.

Updated 2026-05-18 - 6 min read

Matrix applications are really about modelling. The entries in a matrix are not just numbers sitting in a table; they represent relationships, transitions or scores. Specialist focuses especially on dominance matrices and Leslie matrices.

Dominance matrices

A dominance matrix can model a round-robin style competition. A common setup is:

• entry $m_{ij}=1$ if team $i$ beats team $j$
• entry $m_{ij}=0$ otherwise
• diagonal entries are $0$, because a team does not play itself

If draws or point systems are included, the question will define the scoring system.

The useful part is that matrix powers track indirect dominance. The matrix $M$ stores one-step dominance. The matrix $M^2$ stores two-step dominance, because each entry combines paths of the form:

$ \text{team }i\rightarrow \text{middle team}\rightarrow \text{team }j. $

Ranking methods often use a matrix like:

$ M+M^2 $

then add across rows to get each team's total dominance score.

Some questions use weighted dominance. For example, direct wins might count more heavily than indirect two-step dominance, giving a ranking matrix such as:

$ 2M+M^2. $

The weighting rule is not universal; it belongs to the model in the question. Once the rule is defined, the interpretation is the same: row totals measure each competitor's score under that model.

Leslie matrices

A Leslie matrix models an age-structured population over discrete time intervals. The population vector stores the number of individuals in each age class:

$ \mathbf n_t= \begin{pmatrix} n_{1,t}\\ n_{2,t}\\ n_{3,t} \end{pmatrix}. $

A simple Leslie model has the form:

$ \mathbf n_{t+1}=L\mathbf n_t. $

The top row contains fecundity rates, which describe births contributed by each age class. The subdiagonal contains survival rates, which describe how many individuals move into the next age group:

$ L= \begin{pmatrix} f_1&f_2&f_3\\ s_1&0&0\\ 0&s_2&0 \end{pmatrix}. $

To move multiple time steps forward:

$ \mathbf n_t=L^t\mathbf n_0. $

The time step must match the age classes. If the classes are yearly age groups, one multiplication usually means one year. If the classes are five-year groups, one multiplication means five years. The matrix model only makes sense when the time interval and the age categories line up.

Modelling assumptions

Dominance and Leslie models are useful, but they are still simplifications. Dominance rankings depend heavily on the scoring rule chosen. Leslie models often assume closed populations, fixed survival rates and fixed fecundity rates. If the context violates these assumptions, the matrix output should be interpreted cautiously.

QCAA mentions other matrix applications such as Leontief input-output models, Markov chains, area transformations, cryptology, eigenvectors and eigenvalues. For the external exam, the assessable applications are focused on Dominance and Leslie matrices, but the broader list explains why matrix notation is useful: it lets a repeated rule be applied systematically.

For a Leslie matrix, entries should usually be non-negative. Survival rates are normally between $0$ and $1$, while fecundity entries can be greater than $1$. If a calculated population becomes negative, that is a sign the model or data entry has failed.

Leslie models usually assume:

• the population is closed, so there is no migration in or out
• one sex is modelled, often females, because births are linked to fecundity
• fecundity and survival rates stay fixed during the projection
• the age classes match the time step
• the final age class has a clear rule if older individuals remain in the model

Some Leslie matrices grow without bound, some trend downward, and some approach a stable age distribution. The matrix output is not automatically a prediction of the real world; it is the consequence of the assumptions in the model.

Interpreting powers

Matrix powers are not just calculator buttons. In a dominance setting, $M^2$ counts two-step pathways. In a Leslie setting, $L^2\mathbf n_0$ is the population after two time steps. The same algebra has different meaning because the entries represent different things.

Dominance matrix setup choices

A dominance matrix is not always built with only $0$ and $1$. Some models use:

• $2$ for a win
• $1$ for a draw
• $0$ for a loss

The diagonal entries remain $0$ because a team is not compared with itself. The question should define the scoring scheme, so read the row-column convention before calculating.

The reason $M^2$ matters is transitivity. An entry of $M^2$ combines pathways where team $i$ dominates some middle team, and that middle team dominates team $j$. This captures indirect strength. If $M+M^2$ is used, direct and indirect dominance are both contributing to the ranking.

For example, suppose teams $A,B,C,D$ have dominance matrix:

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

Then:

$ M^2= \begin{pmatrix} 0&0&1&2\\ 1&0&0&1\\ 1&0&0&0\\ 0&1&1&0 \end{pmatrix}. $

The entry $(M^2)_{14}=2$ means team $A$ has two two-step dominance pathways to team $D$. For instance, $A$ dominates $B$ and $B$ dominates $D$, and $A$ dominates $C$ and $C$ dominates $D$.

Using $M+M^2$ gives:

$ M+M^2= \begin{pmatrix} 0&1&2&2\\ 1&0&1&2\\ 1&0&0&1\\ 1&1&1&0 \end{pmatrix}. $

Row sums are:

$ 5,\quad4,\quad2,\quad3. $

Under this model, the ranking is $A$, then $B$, then $D$, then $C$. A different weighting, such as $2M+M^2$ or $M+\frac12M^2$, could change the ranking because the model is valuing direct and indirect dominance differently.

Leslie matrix interpretation

The top row of a Leslie matrix is about new births. The subdiagonal is about survival into the next age class. For a four-class model:

$ L= \begin{pmatrix} f_1&f_2&f_3&f_4\\ s_1&0&0&0\\ 0&s_2&0&0\\ 0&0&s_3&0 \end{pmatrix}. $

Multiplying $L\mathbf n_t$ gives the next population vector. The first entry is a birth calculation, while the lower entries are survivors moving down one age category.

Quick check

Sources

• QCAA Specialist Mathematics subject page
• QCAA Specialist Mathematics 2025 syllabus
• Biology LibreTexts: Leslie Matrix Models
• Wolfram MathWorld: Tournament Matrix