IB Mathematics Applications & Interpretation Topic AHL 1.15—eigenvalues and Eigenvectors Questionbank (HL)

IB Mathematics Applications & Interpretation Topic AHL 1.15—eigenvalues and Eigenvectors Questionbank (HL) | RevisionDojo

Practice IB Mathematics Applications & Interpretation (AI) Topic AHL 1.15—eigenvalues and Eigenvectors with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for AHL 1.15—eigenvalues and Eigenvectors and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Question 1

HLPaper 1

Let $z = 1 + i$ . Consider the matrix $A$ given by $A = (z 0 1 \overline{z})$ , where $\overline{z}$ is the complex conjugate of $z$ .

Prove that $z^{2} - 2 z + 2 = 0$ and use this to show that $z^{2} = 2 z - 2$ .

[2]

Determine the exact value of $(z - 1)^{8}$ .

[4]

Verify that $A^{4} = - 4 I$ .

[3]

Show that $A^{- 1} = \frac{1}{2} (2 I - A)$ .

[2]

Question 2

HLPaper 1

Consider the matrix $W= \begin{pmatrix} 4 & -2 \\ -2 & 4 \end{pmatrix}$ .

Find the eigenvectors corresponding to the eigenvalue $λ = 6$ for the matrix $W$ .

[3]

Write down the other eigenvalue of $W$ .

[1]

Determine whether the matrix $W$ is diagonalizable.

[2]

Question 3

HLPaper 1

Find a relationship between $a$ and $b$ if the matrices $M = (12 a 3)$ and $N = (12 b 3)$ commute under matrix multiplication.

[4]

Find the value of $a$ if the determinant of matrix $M$ is $- 1$ .

[2]

Write down $M^{- 1}$ for this value of $a$ .

[1]

Question 4

HLPaper 1

The day-to-day lunch choice of a student is modelled by the transition matrix $T = (0.8 0.2 0.3 0.7)$ , where the states are Cafeteria and Food Truck in that order. The eigenvalues of $T$ are $1$ and $0.5$ . If the student chooses the Cafeteria on one day, the probability of choosing the Food Truck the next day is $0.2$ . If the student chooses the Food Truck, the probability of choosing the Cafeteria the next day is $0.3$ .

Determine an eigenvector associated with the eigenvalue $1$ . Present your result as $(a b)$ , where $a, b \in Z$ .

[3]

Using your result from the previous part, or by another method, calculate the stationary (long-term) probability that the student chooses the Cafeteria. Express your final answer as $\frac{p}{q}$ , where $p, q \in Z^{+}$ .

[2]

Question 5

HLPaper 2

Consider the matrix $A = (2213)$ . Given that $A^{2} + p A + q I = O$ where $p, q \in Z$ and $O$ is the zero matrix.

Determine the scalars $μ$ such that the matrix $(A - μ I)$ has no inverse.

[5]

Find the values of $p$ and $q$ .

[5]

Hence, demonstrate that $I = \frac{1}{4} A (5 I - A)$ .

[4]

Use the expression derived previously to justify why $A$ must be invertible.

[3]

Utilize the values found earlier to express $A^{4}$ in the form $m A + n I$ , where $m, n \in Z$ .

[5]

IB Mathematics Applications & Interpretation AHL 1.15—eigenvalues and Eigenvectors Questionbank

Unlock all 37 questions with a free account

IB Mathematics Applications & Interpretation AHL 1.15—eigenvalues and Eigenvectors Questionbank

IB Mathematics Applications & Interpretation (AI) Topic AHL 1.15—eigenvalues and Eigenvectors Questionbank

Unlock all 37 questions with a free account