Year 12 Discrete Mathematics: Transition matrices

Transition matrices model systems that move from one stage to the next in a fixed way. They are one of the most important modelling tools in Year 12 discrete mathematics because they turn repeated change into matrix multiplication. The main idea is not just numerical calculation. It is interpretation. A transition matrix describes how quantities flow between categories, so rows, columns, and state vectors all have a meaning that must stay visible throughout the working.

Before multiplying anything, decide what the rows represent, what the columns represent, and whether the current state vector should be multiplied on the left or the right.

Subtopic 1: One-step transitions

A one-step transition question asks you to move from one state to the next. The matrix gives the rule, and the state vector gives the current distribution. Once the setup is interpreted correctly, the next state comes from multiplication. The hard part is usually not the arithmetic. It is choosing the correct multiplication order and making sure the meaning of the entries is consistent.

Worked example 1

Problem: A system has state vector

[100, 50]

and transition matrix

[[0.8, 0.3], [0.2, 0.7]]

, where the next state is found by multiplying the matrix by the current column vector. Find the next state.

Write the current state as a column vector: $[[100], [50]]$ .
Multiply: first entry $0.8 (100) + 0.3 (50) = 95$ .
Second entry: $0.2 (100) + 0.7 (50) = 55$ .

Answer: the next state is

[[95], [55]]

Subtopic 2: Repeated transitions

One of the main strengths of transition matrices is that they can model repeated updates efficiently. Two transitions mean multiplying by the matrix twice, which is why powers of the matrix appear naturally. Conceptually, though, the meaning is simple: each multiplication applies the same transition rule one more time. This lets you study how systems evolve across several steps without rebuilding the whole process from scratch each time.

Worked example 2

Problem: Using the same system as above, find the state after two steps.

From the first example, after one step the state is $[[95], [55]]$ .
Multiply again by the same transition matrix.
First entry: $0.8 (95) + 0.3 (55) = 92.5$ .
Second entry: $0.2 (95) + 0.7 (55) = 57.5$ .

Answer: after two steps the state is

[[92.5], [57.5]]

Interpreting the model

Transition-matrix questions often represent movement between groups, locations, or conditions. That means a purely numerical answer is usually incomplete unless it is linked back to the context. If the first component of the state vector counts people in city A and the second counts people in city B, then your answer should be interpreted in those terms. This is also where errors become visible. If a model is meant to preserve a total population but your result changes the total unexpectedly, the setup or the multiplication order may need checking.

Worked example 3

Problem: Explain what the matrix entry

0.3

in the top-right position of the example transition matrix means.

The top-right entry contributes to the first category next step.
It comes from the second category in the current state.
So it represents the proportion of the second category that moves into the first category in one step.

Answer: it means

30 %

of the second category transfers into the first category each step.

Common traps

Multiplying in the wrong order and producing a dimension or interpretation error.
Using the matrix mechanically without stating what each category means.
Forgetting that repeated transitions correspond to matrix powers or repeated multiplication.
Ignoring whether totals or proportions should stay sensible in context.

Revision focus

A strong revision habit is to annotate every transition model with words before doing the arithmetic. Write what each component of the state vector represents and what each row or column of the matrix is doing. This turns multiplication into interpretation and makes it much easier to catch orientation errors early.

It also helps to check the result against common sense. If quantities should stay non-negative, they must. If a closed system should keep the same total, verify that total. In discrete modelling, a small arithmetic error often reveals itself first as an unrealistic interpretation.

Practice links

Level 1: Transition matrices