Applications and Interpretation HL includes several topics that most students have never met in CBSE, ICSE or IGCSE: matrices with eigenvalues, Markov chains, and graph theory. They can feel unfamiliar at first – but they are also very logical and procedural, and with the right GDC skills they become strong scoring areas.
Matrices
- Matrix operations: addition, multiplication, determinants and inverses (2 × 2 by hand, larger with the GDC).
- Solving systems of linear equations using matrices.
- Transformations of the plane using 2 × 2 matrices: rotations, reflections, stretches and enlargements, and the determinant as an area scale factor.
- Eigenvalues and eigenvectors of 2 × 2 matrices, and diagonalisation – used to find powers of a matrix.
Markov chains (transition matrices)
A Markov chain models a system that moves between states with fixed probabilities – for example, customers switching between phone networks each month.
- Build a transition matrix from a description or diagram. Check each column (or row, depending on the convention used) adds to 1.
- Multiply by an initial state to find the distribution after n steps.
- Find the steady state (long-term distribution) – using powers of the matrix or the eigenvector for eigenvalue 1.
The most common error is setting up the matrix the wrong way round. Decide the convention at the start and label rows and columns.
Graph theory
- Vocabulary: vertices, edges, degree, simple and complete graphs, walks, trails, paths and cycles.
- Adjacency matrices, and using powers of them to count walks.
- Minimum spanning trees with Kruskal's and Prim's algorithms.
- The Chinese postman problem (route inspection) and the travelling salesman problem, using upper and lower bounds.
Graph theory questions reward neat, step-by-step working: list edges in the order chosen, show the bound calculations, and state the final answer clearly.
How these topics appear in exams
- A context paragraph (traffic, networks, populations), then a model to build.
- Several calculation parts – often with the GDC.
- An interpretation part: what the result means, or a limitation of the model.
AI HL rewards that final interpretation step. A correct number without a sentence of meaning can lose marks.
GDC skills to master
- Entering matrices, finding inverses and determinants.
- Raising matrices to powers (for Markov chains).
- Solving systems of equations.
- Checking eigenvalue calculations.
A study plan
- Matrices first, by hand for 2 × 2 cases, then with the GDC.
- Transformations with sketches of the unit square.
- Markov chains from worded contexts – setup practice is the key.
- Graph theory algorithms with clear written steps.
- Past papers by topic, then mixed.
How Maths Bodhi helps AI HL students
- A diagnostic to find whether the gap is in setup, GDC use or interpretation.
- Context-to-model drills: turning a paragraph into a matrix or graph.
- GDC sessions on the student's own calculator.
- Interpretation practice with written feedback.
Ajay Vatsyayan (M.Tech, 14+ years teaching maths) teaches IB DP maths at Maths Bodhi and has taught students from schools across Gurgaon, including DPS International, Amity Global School, Lancers International School and Pathways World School. Maths Bodhi is independent and not affiliated with any school. Read Ajay's profile, or see our AI HL tutor page.
Home sessions across Gurgaon – including Golf Course Road and Sector 43 – or online. Book a free demo class.
Related: GDC skills for IB maths, AI SL modelling with functions and the IB DP maths guide for Gurgaon families.
Course content checked against the current IB AI guide by the Maths Bodhi team (October 2026). Study plan reviewed by Ajay Vatsyayan.
Frequently asked questions
- Are matrices in IB maths AI SL?
- No. Matrices, eigenvalues, Markov chains and graph theory are AI HL topics.
- How do you find the steady state of a Markov chain?
- Raise the transition matrix to a high power and multiply by the initial state, or find the eigenvector for eigenvalue 1 and scale it so its entries add to 1.
- Which graph algorithms are in IB maths AI HL?
- Kruskal's and Prim's algorithms for minimum spanning trees, the Chinese postman problem, and the travelling salesman problem with upper and lower bounds.