Matrices and Markov chains

Paper labels are an unofficial, independently authored grouping, applied only where a question's own format genuinely matches a real paper convention (such as IB Mathematics AA's non-calculator/calculator split). They do not reproduce any exam board's real paper numbering or mark allocation, and uncertain questions are labelled general practice instead.

• 1Given A = [[2, 3], [−1, 4]] and B = [[5, −2], [0, 3]], find (a) A + B and (b) A − B.Paper 1 styleEasy2 marksCalculator+
• 2Marking 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.Marking analysisEasy2 marksCalculator+
• 3Given A = [[1, 2], [3, 4]] and B = [[2, 0], [1, 3]], find the matrix product AB.Paper 1 styleEasy3 marksCalculator+
• 4Marking 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.Marking analysisEasy3 marksCalculator+
• 5Find the determinant of the matrix A = [[4, 7], [2, 6]].Paper 1 styleEasy2 marksCalculator+
• 6Marking 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.Marking analysisEasy2 marksCalculator+
• 7Find the inverse of the matrix A = [[3, 5], [1, 2]].Paper 1 styleEasy3 marksCalculator+
• 8Marking 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.Marking analysisEasy3 marksCalculator+
• 9Use a matrix method to solve the simultaneous equations 2x + 3y = 12 and x − y = 1.Paper 2 styleMedium5 marksCalculator+
• 10Marking 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.Marking analysisMedium5 marksCalculator+
• 11Find the determinant of the 3×3 matrix A = [[1, 2, 3], [0, 1, 4], [5, 6, 0]].Paper 2 styleMedium4 marksCalculator+
• 12Marking 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.Marking analysisMedium4 marksCalculator+
• 13Given A = [[1, 2, 3], [4, 5, 6]] and B = [[7, 8], [9, 10], [11, 12]], find the matrix product AB, stating its dimensions.Paper 2 styleMedium4 marksCalculator+
• 14Marking 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.Marking analysisMedium4 marksCalculator+
• 15A 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.Paper 1 styleEasy3 marksCalculator+
• 16Marking 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.Marking analysisEasy3 marksCalculator+
• 17Using 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).Paper 1 styleMedium4 marksCalculator+
• 18Marking 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.Marking analysisMedium4 marksCalculator+
• 19For 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.Paper 2 styleMedium5 marksCalculator+
• 20Marking 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.Marking analysisMedium5 marksCalculator+
• 21For 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.Paper 2 styleMedium4 marksCalculator+
• 22Marking 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.Marking analysisMedium4 marksCalculator+
• 23Solve 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.Paper 2 styleMedium5 marksCalculator+
• 24Marking 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.Marking analysisMedium5 marksCalculator+
• 25A 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.Paper 2 styleMedium5 marksCalculator+
• 26Marking 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.Marking analysisMedium5 marksCalculator+
• 27A 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.Paper 2 styleMedium4 marksCalculator+
• 28Marking 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.Marking analysisMedium4 marksCalculator+
• 29Show 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.Paper 3 style (extended)Medium4 marksCalculator+
• 30Marking 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.Marking analysisMedium4 marksCalculator+
• 31Explain 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.Paper 1 styleEasy3 marksNo calculator+
• 32Marking 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.Marking analysisEasy3 marksNo calculator+