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IB Maths AI HL Ahl 1 15 Hl Eigenvalues and Eigenvectors Questions

Practise finding eigenvalues and eigenvectors, verifying eigenpairs, diagonalising suitable 2×2 matrices and using A^n to interpret long-run models.

Syllabus
First assessment 2021
Course
Mathematics: applications and interpretation HL
Level
HL

Exam points

  • Solve det(A−λI)=0 to find eigenvalues and calculate a corresponding non-zero eigenvector for each.
  • Verify eigenpairs and diagonalise a 2×2 matrix when distinct real eigenvalues permit it.
  • Use A^n=PD^nP⁻¹ to calculate a required term or interpret long-run population or predator–prey behaviour.

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Study AHL 1.15 (HL)—Eigenvalues and eigenvectors conceptsOpen the corresponding Concept lesson for explanations, examples and exam connections.

IB Maths AI HL Ahl 1 15 Hl Eigenvalues and Eigenvectors Questions question 1

[Maximum number: 6]

A geneticist uses a Markov chain model to investigate changes in a specific gene in a cell as it divides. Every time the cell divides, the gene may mutate between its normal state and other states.
The model is of the form

(Xn+1Zn+1)=M(XnZn)\binom{X_{n+1}}{Z_{n+1}}=\boldsymbol{M}\binom{X_n}{Z_n}

where XnX_n is the probability of the gene being in its normal state after dividing for the nth time, and ZnZ_n is the probability of it being in another state after dividing for the nth time, where n∈Nn\in\mathbb{N}.
Matrix M\boldsymbol{M} is found to be (0.94b0.060.98)\left(\begin{smallmatrix}0.94&b\\0.06&0.98\end{smallmatrix}\right).

Question (a)

(a)

Find the eigenvalues of M.

[ 3 ]

Question (b)

(b)

Find the eigenvectors of M.

[ 3 ]
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