Unit 1: Number and Algebra
Topic 1.15: Eigenvalues & eigenvectors (HL only) Questions
Practice 12 exam-style questions for IB Math AI SL Topic 1.15. Review the question stems below, then unlock the full Question Bank to access markschemes, model answers, and AI grading.
11 mark
2026
In a transition (Markov) matrix, the long-run steady state is the eigenvector for:
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2026
To find the eigenvalues of a 2×2 matrix A you solve:
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2026
If Av = λv with v ≠ 0, then v is called:
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2026
The eigenvalues of the diagonal matrix [[6, 0], [0, −2]] are:
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2026
The matrix A = [[2, 0], [3, -1]] has eigenvalues 2 and −1. Find an eigenvector for λ = 2.
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2026
Interpret what this says about repeatedly applying the matrix to a vector.
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2026
For a diagonalised A = PDP⁻¹ with D = [[1, 0], [0, 0.3]], the matrix Dⁿ as n → ∞ tends to:
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2026
Find the eigenvalues of A = [[4, 1], [2, 3]].
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2026
A = [[5, 4], [2, 3]] has eigenvalues 1 and 7 with eigenvectors (1, −1) and (2, 1) respectively. Write down P and D for A = PDP⁻¹, then state D².
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2026
Customers each month stay with brand X or switch to brand Y by the transition matrix T = [[0.9, 0.2], [0.1, 0.8]]. (a) Show that λ = 1 is an eigenvalue. (b) Find the long-run market share of brand X.
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2026
A predator–prey model updates populations each season by M = [[1.2, −0.2], [0.3, 0.9]]. Find the eigenvalues of M, giving answers to 3 s.f.
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2026
A page-rank style network has matrix A = [[0, 0.5], [1, 0.5]] with eigenvalues 1 and −0.5. The corresponding eigenvectors are (1, 2) and (1, −1). Find the long-run distribution and comment on the role of the eigenvalue −0.5.
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