IA idea · Matrices, transformations & Markov chains

Balancing traffic flow through a road network with linear equations

AI HLAI SLAA HL Solid Also in: Networks & graphs, Modelling

Research question

From traffic counts you take at the edges of a small road network, what flows on the inner roads are possible, why is the system underdetermined, and what extra count or rule makes the answer unique?

Adapt it: change the place, the data or the comparison until the question is yours.

Free: the A–E checklist an examiner uses, by email ↓

Why it makes a good exploration

Conservation at each junction gives a linear system you solve with matrices. The underdetermined case is unusual and genuinely interesting: you have to reason about a whole family of solutions.

The mathematics you'll need

Systems of linear equations
Matrix form and row reduction
Underdetermined systems and parameters
Non-negativity constraints on flows
Checking with an extra count

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

Count vehicles entering and leaving a small network (a few junctions near school) over the same time window, safely and from the pavement.

OpenStreetMap — Free map with exact coordinates of schools, hospitals, shops and stations; export or read off coordinates.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Draw the network and label flows.
Count the boundary flows.
Set up and solve the system with a parameter.
Use non-negativity to find the possible range.
Check with an extra count and reflect on counting error.

Pitfalls that cost marks

Counts not taken in the same time window.
Ignoring that flows can't be negative.
No check against an inner road count.

Showing personal engagement

Count your own local junctions.
Predict the busiest inner road.
Model a road closure.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Not a natural fit	The core technique sits in the AI course or at HL; an AA SL student could use it only as clearly explained new mathematics.
AA HL	Good fit	Systems of linear equations; Matrix form and row reduction
AI SL	Good fit	Systems of linear equations; Matrix form and row reduction
AI HL	Good fit	Systems of linear equations; Matrix form and row reduction

Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Personal engagement (C)

Collect the data for your matrix yourself (counting transitions, measuring a shape), and choose the states or the transformation from a situation you care about.

Reflection (D)

Question the model's assumptions: is the process memoryless, are the probabilities constant, does the transformation preserve what it should? Say how each affects your conclusion. For this idea, start with: counts not taken in the same time window — say how it affects your answer.

Use of mathematics (E)

SL: Matrices are in AI HL. At SL, keep to small matrices you explain carefully as new mathematics, with every multiplication shown once and the result interpreted.

HL: Transition matrices, powers, steady states and eigenvalues used correctly, with diagonalisation or a general result derived, and the long-run behaviour interpreted in context.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Use least squares when your counts don't balance exactly.

Extending it for HL

Diagonalise the matrix to find a formula for the nth state, or compare the steady state with what your data actually shows.

See a complete IA, marked

Our annotated exemplar How long does a game of Snakes and Ladders last on my grandmother's board? (AI HL) asks a different question, but shows how a complete matrices & markov exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

Before you start: the checklist an examiner uses

Every check for Criteria A–E in a 4-page PDF, the mistakes that cost the most marks and a self-assessment grid. We'll email it with a short IA tip every few days, timed to your deadline if you give it. Free — no account, no payment.

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).