Mathematics: Applications & Interpretation HL
Matrices and Markov chains — Topic 1 HL
- 1.
Given A = [[2, 3], [−1, 4]] and B = [[5, −2], [0, 3]], find (a) A + B and (b) A − B.
[2 marks] - 2.
Marking analysis: A learner attempts the following task: “Given A = [[2, 3], [−1, 4]] and B = [[5, −2], [0, 3]], find (a) A + B and (b) A − B.” Their response addresses only this point: “Adds corresponding entries to obtain A + B = [[7, 1], [−1, 7]].” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks] - 3.
Given A = [[1, 2], [3, 4]] and B = [[2, 0], [1, 3]], find the matrix product AB.
[3 marks] - 4.
Marking analysis: A learner attempts the following task: “Given A = [[1, 2], [3, 4]] and B = [[2, 0], [1, 3]], find the matrix product AB.” Their response addresses only this point: “Multiplies row 1 of A by each column of B: (1×2 + 2×1, 1×0 + 2×3) = (4, 6).” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 5.
Find the determinant of the matrix A = [[4, 7], [2, 6]].
[2 marks] - 6.
Marking analysis: A learner attempts the following task: “Find the determinant of the matrix A = [[4, 7], [2, 6]].” Their response addresses only this point: “Uses det(A) = ad − bc = (4)(6) − (7)(2).” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks] - 7.
Find the inverse of the matrix A = [[3, 5], [1, 2]].
[3 marks] - 8.
Marking analysis: A learner attempts the following task: “Find the inverse of the matrix A = [[3, 5], [1, 2]].” Their response addresses only this point: “Calculates det(A) = (3)(2) − (5)(1) = 1.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 9.
Use a matrix method to solve the simultaneous equations 2x + 3y = 12 and x − y = 1.
[5 marks] - 10.
Marking analysis: A learner attempts the following task: “Use a matrix method to solve the simultaneous equations 2x + 3y = 12 and x − y = 1.” Their response addresses only this point: “Writes the system in matrix form [[2, 3], [1, −1]][x, y]ᵀ = [12, 1]ᵀ.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[5 marks] - 11.
Find the determinant of the 3×3 matrix A = [[1, 2, 3], [0, 1, 4], [5, 6, 0]].
[4 marks] - 12.
Marking analysis: A learner attempts the following task: “Find the determinant of the 3×3 matrix A = [[1, 2, 3], [0, 1, 4], [5, 6, 0]].” Their response addresses only this point: “Expands along the first row: det(A) = 1(1×0 − 4×6) − 2(0×0 − 4×5) + 3(0×6 − 1×5).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 13.
Given A = [[1, 2, 3], [4, 5, 6]] and B = [[7, 8], [9, 10], [11, 12]], find the matrix product AB, stating its dimensions.
[4 marks] - 14.
Marking analysis: A learner attempts the following task: “Given A = [[1, 2, 3], [4, 5, 6]] and B = [[7, 8], [9, 10], [11, 12]], find the matrix product AB, stating its dimensions.” Their response addresses only this point: “Confirms the product is defined since A is 2×3 and B is 3×2, so AB will be 2×2.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 15.
A machine is always in one of two states: Working (W) or Broken (B). If it is Working on a given day, the probability it is Working the next day is 0.8. If it is Broken, the probability it is Working the next day is 0.3. The machine is Working today. Use the transition matrix P = [[0.8, 0.2], [0.3, 0.7]] and the state vector [1, 0] to find the probability distribution for tomorrow.
[3 marks] - 16.
Marking analysis: A learner attempts the following task: “A machine is always in one of two states: Working (W) or Broken (B). If it is Working on a given day, the probability it is Working the next day is 0.8. If it is Broken, the probability it is Working the next day is 0.3. The machine is Working today. Use the transition matrix P = [[0.8, 0.2], [0.3, 0.7]] and the state vector [1, 0] to find the probability distribution for tomorrow.” Their response addresses only this point: “Multiplies the row state vector by the transition matrix: [1, 0] × P.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 17.
Using the same machine from the previous question (transition matrix P = [[0.8, 0.2], [0.3, 0.7]], starting state [1, 0] for Working today), find the probability distribution for the day after tomorrow (two days ahead).
[4 marks] - 18.
Marking analysis: A learner attempts the following task: “Using the same machine from the previous question (transition matrix P = [[0.8, 0.2], [0.3, 0.7]], starting state [1, 0] for Working today), find the probability distribution for the day after tomorrow (two days ahead).” Their response addresses only this point: “Uses tomorrow's state vector [0.8, 0.2] found previously.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 19.
For the transition matrix P = [[0.8, 0.2], [0.3, 0.7]], find the steady-state (long-run) probability distribution π = [π₁, π₂], using πP = π and π₁ + π₂ = 1.
[5 marks] - 20.
Marking analysis: A learner attempts the following task: “For the transition matrix P = [[0.8, 0.2], [0.3, 0.7]], find the steady-state (long-run) probability distribution π = [π₁, π₂], using πP = π and π₁ + π₂ = 1.” Their response addresses only this point: “Sets up the steady-state equations from πP = π: 0.8π₁ + 0.3π₂ = π₁ and 0.2π₁ + 0.7π₂ = π₂.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[5 marks] - 21.
For the transition matrix P = [[0.8, 0.2], [0.3, 0.7]], calculate P², and explain what the entries of P² represent in the context of the Markov chain.
[4 marks] - 22.
Marking analysis: A learner attempts the following task: “For the transition matrix P = [[0.8, 0.2], [0.3, 0.7]], calculate P², and explain what the entries of P² represent in the context of the Markov chain.” Their response addresses only this point: “Multiplies P by itself to obtain P² = [[0.70, 0.30], [0.45, 0.55]].” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 23.
Solve the matrix equation AX = B for the column vector X, where A = [[2, 1], [1, 1]] and B = [[8], [5]], by finding A⁻¹ and using X = A⁻¹B.
[5 marks] - 24.
Marking analysis: A learner attempts the following task: “Solve the matrix equation AX = B for the column vector X, where A = [[2, 1], [1, 1]] and B = [[8], [5]], by finding A⁻¹ and using X = A⁻¹B.” Their response addresses only this point: “Calculates det(A) = (2)(1) − (1)(1) = 1.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[5 marks] - 25.
A three-state weather model has transition matrix P = [[0.7, 0.2, 0.1], [0.1, 0.6, 0.3], [0.2, 0.2, 0.6]] for states (Sunny, Cloudy, Rainy) respectively. Today's distribution is [0.5, 0.3, 0.2]. Find tomorrow's probability distribution.
[5 marks] - 26.
Marking analysis: A learner attempts the following task: “A three-state weather model has transition matrix P = [[0.7, 0.2, 0.1], [0.1, 0.6, 0.3], [0.2, 0.2, 0.6]] for states (Sunny, Cloudy, Rainy) respectively. Today's distribution is [0.5, 0.3, 0.2]. Find tomorrow's probability distribution.” Their response addresses only this point: “Multiplies the row state vector [0.5, 0.3, 0.2] by the transition matrix P.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[5 marks] - 27.
A city's residents are classified as living Downtown (D) or in the Suburbs (S). Each year, 40% of Downtown residents move to the Suburbs, and 50% of Suburb residents move Downtown. The transition matrix is P = [[0.6, 0.4], [0.5, 0.5]]. If the population starts entirely Downtown, [1, 0], find the distribution after 2 years.
[4 marks] - 28.
Marking analysis: A learner attempts the following task: “A city's residents are classified as living Downtown (D) or in the Suburbs (S). Each year, 40% of Downtown residents move to the Suburbs, and 50% of Suburb residents move Downtown. The transition matrix is P = [[0.6, 0.4], [0.5, 0.5]]. If the population starts entirely Downtown, [1, 0], find the distribution after 2 years.” Their response addresses only this point: “Multiplies [1, 0] by P to obtain the distribution after 1 year: [0.6, 0.4].” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 29.
Show that the uniform distribution π = [1/3, 1/3, 1/3] is the steady-state distribution for the transition matrix P = [[0.7, 0.2, 0.1], [0.1, 0.6, 0.3], [0.2, 0.2, 0.6]] from an earlier question, by verifying that πP = π.
[4 marks] - 30.
Marking analysis: A learner attempts the following task: “Show that the uniform distribution π = [1/3, 1/3, 1/3] is the steady-state distribution for the transition matrix P = [[0.7, 0.2, 0.1], [0.1, 0.6, 0.3], [0.2, 0.2, 0.6]] from an earlier question, by verifying that πP = π.” Their response addresses only this point: “Multiplies π = [1/3, 1/3, 1/3] by P.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 31.
Explain what it means for a Markov chain to be 'regular', and explain why a regular Markov chain always converges to a unique steady-state distribution regardless of its starting state.
[3 marks] · no calculator - 32.
Marking analysis: A learner attempts the following task: “Explain what it means for a Markov chain to be 'regular', and explain why a regular Markov chain always converges to a unique steady-state distribution regardless of its starting state.” Their response addresses only this point: “Explains that a Markov chain is regular if some power of its transition matrix has all strictly positive entries, meaning every state can eventually be reached from every other state.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] · no calculator