VCE Further Mathematics / General Mathematics, Units 3 & 4 — exams 2006–2026

Built from the local corpus at corpus/text\*.txt (128 files: every VCAA Further Mathematics / General Mathematics exam paper and examination report / assessment guide 2006–2026, including NHT papers and the March 2023 sample papers).

Every quotation below is verbatim from a corpus file. Where the PDF→text conversion mangled a matrix layout, only the prose stem is quoted. The 2011 Exam 1 and Exam 2 files are stored in a font-shifted encoding (+29 ASCII); they were decoded before quoting, which collapses inter-word spaces — 2011 quotes are therefore reconstructed word spacing, flagged inline.

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0. Structural map — where Matrices lives, and how much it is worth

This matters because it determines the shape of every question type below.

Sources: structure tables in Documents_exams_mathematics_2013_2013furmath1-w.txt (13+27), ...2013furmath2-w.txt (15 + 3×15), ...2016_2016furmath1-w.txt (24+16), ...2016_2016furmath2-w.txt (36 + 2×12), ...2020_2020furmath1-w.txt (30+10), ...2020_2020furmath2-w.txt (45 + 1×15), 2025-04_genmath-specs-w.txt lines 20–28 ("8 will be allocated to matrices" / "12 marks will be allocated to matrices"), Documents_exams_mathematics_2023_2023genmath2-w.txt (14 questions, 60 marks).

Question numbering of the Matrices block in Exam 2, 2023+:

Formula sheet. From 2023 the General Mathematics formula sheet carries, under "Matrices", only four lines (Documents_exams_mathematics_generalmaths2-formula-w.txt, 2025-11_2025-GeneralMaths2.txt):

determinant of a 2 × 2 matrix       A = [a b; c d],  det A = |a b; c d| = ad − bc
inverse of a 2 × 2 matrix           A⁻¹ = (1/det A)[d −b; −c a],  where det A ≠ 0
recurrence relation                 S0 = initial state,  S(n+1) = T Sn + B
Leslie matrix recurrence relation   S0 = initial state,  S(n+1) = L Sn

The 2006–2022 Further Mathematics formula sheet carried only the first two lines. The explicit S(n+1) = T Sn + B and the Leslie line are new to the 2023+ study design — this is the single biggest content shift in the topic and drives the new question types in §1.11 and §1.12.

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1. COMPLETE QUESTION-TYPE CATALOGUE

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1.1 Matrix order / size; element notation aᵢⱼ; identifying elements from context

1.1a "Write down the order of matrix X" / "What is the order of matrix X?"

What it looks like. A labelled matrix of real-world data is given (prices, enrolments, populations). The very first part of the first Exam 2 matrices question asks for its order. It is the single most reliable question in the whole topic: it appears as part (a) of Question 1 in almost every Exam 2 paper from 2006 to 2021, and returns in 2025/2026 NHT.

Marks: always 1 mark. Presence: Exam 2 gimme (~20 of 21 papers); Exam 1 appears as order-of-a-product instead (see 1.1c).

Success rates are very high — 2016: 91 %; 2019: 95 %; 2020: 95 %; 2021: 93 %; 2025: 96 %.

Real examples:

• 2006 Exam 2 Question 1a — "a. Write down the order of matrix Q." (1 mark)
• 2009 Exam 2 Question 1a — "a. What is the order of matrix P?" (1 mark) — answer 2 × 3
• 2014 Exam 2 Question 1a — "a. Write down the order of matrix V." (1 mark) — answer 4 × 2; report: "Some students reversed the two numbers, writing 2 × 4, which was not accepted."
• 2016 Exam 2 Question 1a — "a. Write down the order of matrix C." → 4 × 1; report: "A common incorrect answer was 1 × 4. An answer of (4, 1) was not accepted."
• 2019 Exam 2 Question 1a — "a. What is the order of matrix Q?" → 3 × 2
• 2020 Exam 2 Question 1a — "a. Write down the order of matrix V." → 1 × 3
• 2021 Exam 2 Question 1a — "a. What is the order of matrix J?" → 3 × 1
• 2025 Exam 2 Question 11a — "a. State the order of matrix E. / Write your answer in the boxes provided. ☐ × ☐" (1 mark) → 3 × 5
• 2026 NHT Exam 2 Question 9a — "a. Write down the order of matrix V in the boxes below. ☐ × ☐"

Note the 2025+ move to boxes (☐ × ☐), which removes the "(4, 1)" failure mode but keeps the rows-before-columns trap.

1.1b Element notation aᵢⱼ — reading an element off a matrix

What it looks like. "The element in row i and column j of matrix M is mᵢⱼ. What does the element m₁₂ indicate?" — or the reverse, "Which element shows the cost for one child ticket?"

Marks: 1 mark. Presence: heavily in both exams. In Exam 1 it is usually a construct-the-matrix- from-a-rule question (1.1d); in Exam 2 it is an interpret-in-context question (see §2.2 for the required answer form).

• 2017 Exam 2 Question 1b — "The element in row i and column j of matrix M is mᵢⱼ. / b. What does the element m₁₂ indicate?" (1 mark). Report answer: "The number of rolls sold in week 1, which was 24." "Some students simply gave the element 24." (i.e. the number alone was insufficient — 75 % scored.)
• 2020 Exam 2 Question 1cii — "The element in row i and column j of matrix Q is qᵢⱼ. / What does the element q₂₃ represent?" Report answer: "The number of shoppers (594) in the clothing area of Westmall (at 1.00 pm)."
• 2023 Exam 2 Question 8a — "The element in row i and column j of matrix N is nᵢⱼ. / a. Which element shows the cost for one child ticket?" (1 mark) → n₃₁
• 2022 NHT Exam 2 Question 1a — "a. Interpret the value of element c₃₁." → "The cost of an evening ticket is $7.00."
• 2024 NHT Exam 2 Question 10 stem — "The '1' at element h₂₅ indicates that barn J can directly communicate to barn M. / The '0' at element h₄₂ indicates that barn L cannot directly communicate to barn J."
• 2022 Exam 1 Question 1 — "The element in row i and column j in matrix B is bᵢⱼ. … The daily cost of renting an adult mountain bike is shown in element"
• 2023 Exam 1 Question 25 — "Element m₂₁ indicates that"
• 2015 Exam 1 Question 1 — "The element in row i and column j of matrix B is bᵢⱼ. The element b₃₂ is the number of"

1.1c Order of a product / whether a product is defined

What it looks like. Pure Exam 1. Given orders (or the matrices), identify the order of a product, the order of an unknown factor, or which of five expressions is defined.

Marks: 1 mark, Exam 1 only (Exam 2 asks this as "Explain why … is not defined", see §1.3b).

• 2006 Exam 1 Question 6 — "If A = [1 3; 6 4; 0 0] and the matrix product XA = [4 1; 1 4; 3 5], then the order of matrix X is" (answer E: 3 × 3). Only 23 % correct — the worst matrices question of 2006. Report solution: "matrix X must have three rows for the product matrix XA … to have three rows / matrix X must have three columns for the product XA to be defined. Thus X must be a (3 × 3) matrix."
• 2007 Exam 1 Question 9 — "Matrix M is a 3 × 4 matrix. Matrix P has five rows. N is another matrix. If the matrix product M(NP) = … then the order of matrix N is"
• 2010 Exam 1 Question 8 — "m and n are positive whole numbers. Matrix P is of order m × n. Matrix Q is of order n × m. The matrix products PQ and QP are both defined"
• 2013 Exam 1 Question 9 — "P, Q, R and S are matrices such that the matrix product P = QRS is defined. Matrix Q and matrix S are square, non-zero matrices for which Q + S is not defined. Which one of the following matrix expressions is defined?"
• 2015 Exam 1 Question 2 — "Which one of the following matrix products is not defined?"
• 2019 Exam 1 Question 1 — "Consider the following four matrix expressions. … How many of these four matrix expressions are defined?"
• 2020 Exam 1 Question 8 / Sample Exam 1 Question 30 — "The transpose of matrix A, for example, is written as Aᵀ. What is the order of the product Cᵀ × (Aᵀ × B)ᵀ?"
• 2021 Exam 1 Question 5 — "A is a 7 × 7 matrix. B is a 10 × 7 matrix. Which one of the following matrix expressions is defined?" (report: "Bᵀ is the transpose of matrix B. Bᵀ is a 7 × 10 matrix, therefore A(Bᵀ) is defined.")
• 2024 NHT Exam 1 Question 29 — "Matrix J is a row matrix of order 1 × n. Matrix K is a column matrix of order n × 1. … How many of the above matrix products are defined?"
• 2025 Exam 1 Question 31 — "Which one of the following computations is defined?"
• 2026 NHT Exam 1 Question 26 — "This calculation will be defined if the order of G is"

1.1d Build a matrix from a rule aᵢⱼ = f(i, j)

Exam 1 only. A near-annual fixture since 2014.

• 2014 Exam 1 Question 6 — "The order of matrix X is 3 × 2. The element in row i and column j of matrix X is xᵢⱼ and it is determined by the rule xᵢⱼ = i + j. The matrix X is"
• 2015 Exam 1 Question 8 — "The order of matrix X is 2 × 3. The element in row i and column j of matrix X is xᵢⱼ and it is determined by the rule xᵢⱼ = i × j."
• 2016 Exam 1 Question 5 — "Let M = [1 2 3 4; 3 4 5 6]. The element in row i and column j of M is mᵢⱼ. The elements of M are determined by the rule"
• 2017 NHT Exam 1 Question 4 — "The elements in matrix M are determined using the rule mᵢⱼ = 2i + j. Matrix M could not be"
• 2019 Exam 1 Question 3 / Sample Exam 1 Question 28 — "The element in row i and column j of matrix P is pᵢⱼ. The elements in matrix P are determined by the rule"
• 2020 Exam 1 Question 6 — "M is a 3 × 3 matrix. It is constructed using the rule mᵢⱼ = 3i + 2j."
• 2023 Exam 1 Question 29 — "Matrix K is a 3 × 2 matrix. The elements of K are determined by the rule kᵢⱼ = (i − j)². Matrix K is"
• 2024 NHT Exam 1 Question 28 — "Matrix D is a 2 × 2 matrix where each element is given by dᵢⱼ. Which rule will result in a binary matrix?"
• 2025 Exam 1 Question 30 — "F is a 4 × 4 matrix. The element in row i and column j of F is fᵢⱼ. The elements are determined by the rule fᵢⱼ = i² − j. How many of the elements in F will be negative?"
• 2017 Exam 1 Question 6 — two rules combined: report solution "aᵢⱼ + bᵢⱼ = 2i + j + i − j = 3i."

1.1e Transpose

• 2016 Exam 1 Question 1 — "The transpose of [2 7 10; 13 19 8] is"
• 2019 NHT Exam 1 Question 3 — "In matrix A, the element a₂₁ = 7. The order of matrix A is 3 × 2. Matrix B is the transpose of matrix A. Matrix B could be"
• 2021 Exam 1 Question 1 — "If matrix M = [3 2; 8 9; 13 7], then its transpose, Mᵀ, is"
• 2019 Exam 2 Question 1d–e — "d. Matrix Rᵀ is the transpose of matrix R. / Complete the matrix Rᵀ below." (1 mark) then "e. Explain what the element in row 3, column 2 of matrix Rᵀ represents." (1 mark)
• 2024 GM Exam 2 Question 9a — assessment-guide video transcript (2025-04_General_Maths_Examination_2.txt): "The majority of students understood the concept of the transpose matrix, which is to transpose the row column designation, so that, for example, element R 1,3 becomes element R 3,1 in the transpose matrix."
• 2026 NHT Exam 2 Question 10b — "R = QᵀM / Calculate matrix R below."
• 2023 GM Sample Exam 2 Question 13b — "The transpose of the permutation matrix, Pᵀ, was used to create a new roster for March."

1.1f Named matrix types (2023+ only)

New in the General Mathematics era: questions that simply test vocabulary.

• 2020 Exam 1 Question 1 — "The matrix [1 0 0; 0 1 1; 1 0 1] is an example of" — A. a binary matrix. B. an identity matrix. C. a triangular matrix. D. a symmetric matrix. E. a permutation matrix.
• 2023 NHT Exam 1 Question 1 — "Which one of the following is not an example of a square matrix?" (options: state / inverse / transition / permutation / communication matrix)
• 2021 NHT Exam 1 Question 2 — "How many of these matrices are diagonal matrices?"
• 2025 Exam 1 Question 25 — "Consider the matrix G where G = [0 1 0; 1 0 1; 0 0 0]. Which one of the following correctly describes matrix G?" (83 % correct). Assessment guide reasoning verbatim: "Binary matrix only contains 1s and 0s / Permutation matrix has one 1 in each row and column. Eliminate B. / Identity matrix only has 1s on the main diagonal and 0s elsewhere. Eliminate C. / Diagonal matrix has all entries outside the main diagonal as 0. Eliminate D."
• 2025 NHT Exam 1 Question 25 — symmetric matrix: "In a symmetric matrix, the elements above the leading diagonal are a mirror image of the elements below the diagonal."
• 2018 NHT Exam 1 Question 1 — "The matrix [1 0 0; 5 1 0; 8 5 1] is an example of a" (answer: triangular)

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1.2 Addition, subtraction, scalar multiplication

1.2a Bare arithmetic (Exam 1 opener)

Almost always Question 1 or 2 of the Exam 1 module, 85–98 % correct.

• 2007 Exam 1 Question 1 — "The matrix sum [0 −4; 2 5] + [5 4; −2 2] is equal to"
• 2008 Exam 1 Question 1 — "If [1 0; 0 1] + [3 7; 8 d] = [4 7; 8 11] then d is equal to"
• 2009 Exam 1 Question 1 — "3[0 2; 1 3] − 2[2 1; 0 7] equals"
• 2012 Exam 1 Question 1 — "2 × [2 8; 4 −1; 3 5] − [3 7; 4 2; 2 3] equals"
• 2017 NHT Exam 1 Question 1 — "[4 2; 1 5] + 3[4 2; 1 5] is equal to"
• 2018 NHT Exam 1 Question 2 — "2 × [3 0; 4 −1] + W = [6 2; 7 0] Which one of the following is matrix W?"
• 2021 NHT Exam 1 Question 1 — "The matrix expression [2 6; 4 1] + [2 6; 4 1] + [4 12; 8 2] − [8 24; 16 2] is equal to"

1.2b Scalar as a percentage change — "Write down the value of the scalar k"

This is the highest-yield scalar question type and it recurs relentlessly. A percentage increase/decrease must be converted to a multiplier.

Marks: 1 mark. Presence: Exam 2 (write the scalar) and Exam 1 (choose the calculation).

• 2021 Exam 2 Question 1b — "b. Elena expected that in February 2021 the sales of all three brands of olive oil would increase by 5 %. / She multiplied matrix J by a scalar value, k, to determine the expected volume of sales for February. / Write down the value of the scalar k. / k = ____" (1 mark). Only 48 % correct. Report: "The correct answer was 1.05. This question was not answered well. Students often responded with 0.05."
• 2018 NHT Exam 2 Question 1d — "The number of farmers who attended the 2018 conference increased by 25 % for each district from the previous year. / Complete the product below with a scalar so that the product gives the number of farmers from each district who attended the 2018 conference. / F₂₀₁₈ = ____ × F₂₀₁₇" → 1.25
• 2022 Exam 2 Question 1a — "Complete the matrix equation below to show the cost of staying at each type of accommodation for five nights. / ____ × [80 140 270] =" → scalar 5
• 2022 NHT Exam 2 Question 1c — "The transport company offers a discount of 20 % on Mondays. / In the box below, complete the expression by writing down the scalar value that will calculate the discount of 20 %." → 0.8
• 2026 NHT Exam 2 Question 10c — "New customers receive a 12 % discount … Matrix M can be multiplied by a scalar, p, to determine the discounted price of the toys. / Write down the value of the scalar p below." Assessment guide: "(100 − 12) % = 88 %" → 0.88
• 2024 GM Exam 1 Question 26 — report reasoning: "A discount of 15 % per annum is equal to multiplying by (100 − 15) % = 85 % = 0.85"
• 2022 Exam 1 Question 2 — "On Sundays, the business increases the daily rental price for each type of bike by 10 %. To determine the rental cost for each type of bike on a Sunday, which one of the following matrix calculations needs to be completed?"
• 2023 NHT Exam 1 Question 4 — "A cinema increased the cost of a child's ticket, an adult's ticket and a senior's ticket by the same scalar factor, a."

1.2c Diagonal matrix as a different percentage change per row

The variant: when each row changes by a different percentage, a scalar no longer works — a diagonal matrix is required. This is a reliably low-scoring question.

• 2018 Exam 2 Question 2a–b — "Annual change: main town increase by 4 %, villages decrease by 1 %, rural areas decrease by 2 % … b. The expected population in each of the three locations in 2019 can be determined from the matrix product P₂₀₁₉ = F × P₂₀₁₈ where F is a diagonal matrix. / Write down matrix F." 40 % correct. Report: "1.04 0 0 / 0 0.99 0 / 0 0 0.98 — Most students recognised that a 3 × 3 matrix was required. Numbers such as 0.04, −0.01 and −0.02 were often included in students' answers."
• 2020 Exam 2 Question 1e — "food increase by 5 %, clothing decrease by 15 %, merchandise decrease by 1 % … The average daily amount, in dollars, expected to be spent in each area in 2020 can be determined by forming the matrix product A₂₀₂₀ = K × A₂₀₁₉ / Write down matrix K." Only 24 % correct. Report: "Most students did not recognise the need to form a 3 × 3 matrix and give a row or column matrix. Many students also were not able to obtain the correct multiplying factors from the percentage changes."
• 2025 Exam 2 Question 11c — "the capacity of the Nursery room will be increased by 25 % and the capacity of the Toddler room will be increased by 50 %. The capacity of the Pre-kinder room will be reduced by 10 %. … Cnew = FC where F is a diagonal matrix. / Write down the matrix F." 44 % correct. Report: "Many responses indicated a poor understanding of what a diagonal matrix is."
• 2006 Exam 1 Question 5 — the Exam 1 form of the same idea: "A new price matrix … can be generated from P by forming a matrix product with the matrix M, where M = [1.2 0; 0 1.35]."

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1.3 Matrix multiplication — validity, computing, interpreting an element of a product

1.3a Compute a product and state what it means (the canonical Exam 2 Q1 pair)

What it looks like. Part (i) "Evaluate the matrix product …"; part (ii) "What does matrix J represent?" / "Explain what the matrix product WN represents."

Marks: 1 + 1. Presence: the most common Exam 2 matrices opener 2006–2021. The interpret half is consistently much harder than the compute half (2016: 79 % compute vs 42 % interpret).

• 2006 Exam 2 Question 1b — "b. i. Evaluate the matrix M, where M = QP. / ii. What information does the elements of matrix M provide?" Report: "Total revenue from selling products A, B and C at Eastown and Noxland"
• 2007 Exam 2 Question 1b — "i. Evaluate the matrix product AB. / ii. Determine the order of matrix product BA. / iii. Explain the information that the matrix product AB provides." Report: "The total energy content of the servings of these four foods in one sandwich."
• 2008 Exam 2 Question 1b — "Let the matrix R = NP. / i. Evaluate the matrix R. / ii. Explain what the matrix element R₂₄ represents." → "The number of students who are expected to get a D in Chemistry"
• 2016 Exam 2 Question 1b — "b. i. Calculate the matrix product J = G × C. / ii. What does matrix J represent?" → [6000]; "Total booking fees collected for the month"
• 2017 NHT Exam 2 Question 1b — "i. Determine the matrix product WN. / ii. Explain what the matrix product WN represents." → [21 200]; "Total cost of all seats in the theatre when every seat is sold."
• 2021 NHT Exam 2 Question 1b — "Let the matrix R = N × E. / i. Determine matrix R. / ii. Explain what the matrix element r₃₂ represents." → "The number of Year 9 students awarded a credit in English."
• 2010 Exam 2 Question 1b–c — "b. Matrix P is found by multiplying matrix N with matrix G so that P = N × G. Evaluate matrix P. / c. In this context, what does the information in matrix P provide?" → [26]; "The total of all the points scored by Oscar in this game"
• 2011 Exam 2 Question 2d (decoded) — "In the context of the problem what information does matrix M contain?" → "Total number of birds, lizards and frogs that were killed"

1.3b "Explain why … is not defined"

Marks: 1 mark. Exam 2. Requires an explicit columns-of-first = rows-of-second statement.

• 2006 Exam 2 Question 1c — "c. Explain why the matrix PQ is not defined." Report answer: "number of columns in P ≠ number of rows in Q", plus the crucial rider: "Many answers illustrated a poor understanding of the need for the number of columns in matrix P to be the same as the number of rows in matrix Q. Some students simply said that the 'second number' in matrix P had to equal the 'first number' in matrix Q. Although this was accepted in 2006, it did not indicate whether the student knew which numbers referred to rows and which to columns."
• 2014 Exam 2 Question 1d — the positive form: "d. Explain, in terms of rows and columns, why the matrix product V × P is defined." → "Number of columns in V = number of rows in P"
• 2026 NHT Exam 2 Question 10a — "a. Explain why MQ is not defined." Assessment guide, with an explicit marking condition: "MQ Is NOT defined because the number of columns in M (1) is not equal to the number of rows in Q (3). | Must reference M & Q OR 1 and 3 for columns/rows"
• 2018 NHT Exam 2 Question 3b — "b. Explain why this equation cannot be solved using the matrix inverse method." Report: "A 3 × 3 matrix is required to find the inverse. Other acceptable answers described the need for a square matrix to find the inverse or that three equations were needed to solve for three unknowns."
• 2006 Exam 1 Question 2 — "Using these matrices, the matrix product that is not defined is"

1.3c Interpret a specific element of a product / a specific calculation

• 2012 Exam 1 Question 2 — "If A = [8 1; 4 2] and B = [3 12; 6 0], then the matrix AB = [30 96; 24 48]. The element '24' in the matrix AB is correctly obtained by calculating"
• 2020 Exam 1 Question 2 — "Matrix Q = A × B. The element in row i and column j of matrix Q is qᵢⱼ. Element q₄₁ is determined by the calculation"
• 2025 Exam 1 Question 26 — "Consider the matrices A, B and C where A = [2 0; 1 6], B = [3; 5] and C = AB. The calculation that correctly determines element c₂₁ is" (84 % correct). Report: "Row 2 × column 1, 1 × 3 + 6 × 5"
• 2021 NHT Exam 1 Question 6 — "The matrix product V × W produces a new matrix, L. The elements in matrix L show the total cost of"
• 2023 NHT Exam 2 Question 1b — "b. Explain what the value of p₁₁ − p₁₂ represents in matrix P." Report: "The difference in the cost of hiring a coach for eight hours compared to hiring a coach for one day."

1.3d Build the multiplier matrix so a stated product works

Given a target product, write down the matrix that produces it. Usually a row/column matrix of counts.

• 2018 Exam 2 Question 1c — "One day Kim travels once on section E and twice on section G. / His total toll cost for this day can be found by the matrix product M × C. / Write down the matrix M." 66 %. Report: [1 0 2]; "Some students gave the correct numbers but in a column matrix."
• 2019 Exam 2 Question 1c — "The total parking fee, in dollars, collected from these 110 parked cars if they were parked for one hour is calculated as follows. P × L = [207.00] where matrix L is a 3 × 1 matrix. Write down matrix L."
• 2022 Exam 2 Question 1b — "Write down the column matrix A for which the product matrix CA gives the total accommodation cost for the five nights."
• 2017 Exam 2 Question 1cii — "Matrix L in this equation is of order 1 × 3. / Write down matrix L." 28 % correct. Report: "Some were correct but many students gave a 3 × 1 matrix." (The report also notes the stem's own introduction was misworded that year.)
• 2023 Exam 2 Question 8c — "The owners of the circus want a 3 × 1 product matrix that displays the revenue for each ticket type: family, adult and child. / This product matrix can be achieved by completing the following matrix multiplication. K × N = [7344; 2430; 1408] / Write down matrix K in the space below."
• 2019 NHT Exam 2 Question 1d — a 4 × 3 selector matrix defined by four element descriptions: "element v₁₁ = total cost of uniforms for all female athletes from Gillen … Complete matrix Q with the missing values."
• 2022 NHT Exam 2 Question 1c — "The total value of each ticket type sold on this day could be determined by multiplying a 3 × 3 matrix called matrix R by matrix C. / Complete matrix R below."
• 2021 NHT Exam 1 Question 7 / 2022 Exam 1 Question 8 — Exam 1 form: pick the product that produces a required 3 × 1 or 1 × 3 result. 2022 report distinguishes all five options explicitly (see §3.3).

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1.4 Using a matrix product to sum rows/columns (the "ones matrix") and cost/revenue applications

This is the type the brief calls the classic product-with-ones-matrix question. It has two faces: (a) the "summing matrix" of 1s, and (b) a price/cost row or column matrix. Both are asked as "Write down a matrix calculation / expression that …".

Marks: 1 mark (occasionally 2 when a numerical answer is also required). Presence: both exams; every single year in some form.

1.4a The summing matrix of ones

• 2012 Exam 2 Question 1c–d — "c. Evaluate the matrix product G = KF, where K = [1 1 1 1]. / d. In the context of the problem, what information does matrix G contain?" The report gives the canonical definition: "The matrix G lists, for each city, the total number of direct flight connections from that city to another city in the network. It refers specifically to the number of direct flights out of each of the four cities to another city in the network, not the inward flights or just 'connections'. The matrix K = [1 1 1 1] is an example of a 'summing matrix' that finds the sum of each column of the 4 × 4 matrix labelled F."
• 2025 Exam 2 Question 11b — "The following matrix multiplication has been completed to determine a new matrix, W. [1 1 1] × E = W / What information does matrix W provide regarding enrolments at the early learning centre?" 57 % correct. Report: "Total enrolments each day of the week"; "Many students gave an answer that suggested that the rows had been summed and not the columns."
• 2026 NHT Exam 2 Question 9b — "Matrix S and matrix V are multiplied to form the matrix product SV. This matrix SV is a row matrix showing the total number of visitors on each day of the week over the three weeks. Write matrix S below." (→ [1 1 1])
• 2023 Exam 2 Question 9b — an averaging/scaling hybrid: "[t] × R × [1;1;1;1;1] / b. Determine the value of t if the circus would like to increase its revenue from ticket sales by 25 %."
• 2023 Exam 2 Question 9a — the averaging variant: "Complete the matrix equation below that calculates the average ticket sales per performance at each of the five locations. _ × R = _" (scalar 1/20)
• 2024 NHT Exam 2 Question 11a — "Complete the following matrix equation that will calculate the average egg weight in a carton for each egg size. _ × [504 600 696 804] = _"
• 2024 NHT Exam 2 Question 11b — "In the space below, write a matrix calculation using Matrix Q and a column matrix to show that the total number of eggs sold on this day was 4824."
• 2013 Exam 2 Question 1a–b — row-sum by hand, then interpret: "a. Find the sum of the elements in row 3 of matrix W. / b. In terms of the breeding ponds described, what does the sum of the elements in row 3 of matrix W represent?" → "Two ponds connect directly by pipe to pond R."
• 2014 Exam 2 Question 1c — "c. In terms of the population of the city, what does the sum of the elements in the second column of matrix V represent?" → "Total number of adult females living in this small city"
• Exam 1 versions: 2010 Q4 ("Which matrix expression results in a matrix that contains the sum of the numbers 2, 5, 4, 1 and 8?"), 2012 Q6, 2013 Q7 ("…generate a matrix that displays the mean mark obtained for each class"), 2015 Q4, 2016 Q4, 2022 NHT Q1 ("Calculating the matrix product MN will produce a matrix that … sums the rows / sums the columns of matrix M"), 2023 NHT Q7, 2026 NHT Q28.

1.4b Cost / revenue: "Write a matrix calculation … to show that the total … is $X"

The dominant modern phrasing. The target value is given; the student must supply the expression.

• 2008 Exam 2 Question 1c — "i. Write down a clearly labelled row matrix, called F, that lists these fees. / ii. Show a matrix calculation that will give the total laboratory fees, L, paid in dollars by the students enrolled in Biology and Chemistry. Find this amount." Report: "Despite a row matrix being required, many students wrote a column matrix instead. The question expected students to label the matrix as shown."
• 2015 Exam 2 Question 1c — "c. i. Write down a matrix calculation in terms of Q and C that results in a matrix that lists the amount paid to each teacher each week." Report: "C × Q"; "Some students incorrectly gave Q × C."
• 2020 Exam 2 Question 1d — "d. Write a matrix calculation, using matrix A₂₀₁₉, showing that the total amount spent by all these shoppers is $9663.20" 60 % correct. Report: "Many students gave the correct matrix calculation. Some had the two matrices but in the incorrect order."
• 2021 NHT Exam 2 Question 1c — "Write down a matrix calculation that determines the total printing cost for distinction (D) certificates."
• 2022 NHT Exam 2 Question 1b — "b. Write a matrix calculation, using matrix C and a row matrix, to show that the total value of the three ticket types sold on this day was $771.50"
• 2019 NHT Exam 2 Question 1cii — "Write a matrix calculation, that includes matrix C, to show that the total cost of uniforms for the event named in part c.i. is contained in the matrix answer of [1030]." Report answer: [2 0 0] × [515; 550; 580] = 1030
• 2019 Exam 2 Question 1b — the arithmetic-expression variant: "b. Write down a calculation to show that 110 parking spaces are full at 1 pm." Report: "The answer is 50 + 20 + 40. Although a simple arithmetic expression was all that was required, a matrix calculation was also accepted."
• 2023 Exam 2 Question 8b — "Complete the matrix equation below to show that purchasing a family ticket could give families a saving of $10. [0 2 2] × N − ____ × N = 10"
• Exam 1: 2008 Q2 ("A matrix product to calculate the total cost of these items is"), 2014 Q5, 2016 Q4, 2017 NHT Q2, 2019 Q2, 2018 NHT Q5, 2023 NHT Q7, 2026 NHT Q28.

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1.5 Identity, inverse, determinant; when an inverse does not exist

Marks: 1 mark in Exam 1; 1–2 marks in Exam 2 (often chained into a simultaneous-equations part). Presence: near-annual in Exam 1; sporadic but recurring in Exam 2 (2006, 2018 NHT, 2020, 2022, 2023 NHT).

1.5a Determinant = 0 ⇒ no inverse ⇒ no unique solution

• 2020 Exam 2 Question 2a — "The value of the determinant of the 2 × 2 matrix is 1. / Use this information to explain why this matrix has an inverse." Only 37 % correct. Report: "The determinant is not equal to zero. Many students did not recognise that a non-zero determinant is required for an inverse matrix to exist. Some stated that the determinant had to be greater than 0."
• 2020 Exam 2 Question 2b — "Write the three missing values of the inverse matrix that can be used to solve these equations." Report: "Some students did not include the negative signs, and others did not correctly interchange the positions of the 5 and 7."
• 2022 Exam 2 Question 2b — "i. Show that the determinant of the matrix [20 40; 30 50] is −200. / ii. Interpret the value of this determinant in relation to the solution of the simultaneous linear equations above." Part ii: 21 % correct. Report: "A unique solution exists, as the determinant is not zero. This question was not answered well, with few students able to demonstrate how to interpret the determinant value. The response also needed both parts."
• 2006 Exam 2 Question 3b–c — "b. Do the equations have a unique solution? Provide an explanation to justify your response. / c. Write down an inverse matrix that can be used to solve these equations." Report: "There is a unique solution since det [2 1 1; 1 −1 1; 0 2 −1] = 1 ≠ 0. Incorrect answers included those where students justified their response by quoting the actual solutions to the equations."
• 2023 NHT Exam 2 Question 3d — determinant supplied to force a missing element: "Given that det [30 8; 22 f] = 304, what is the weekday hourly rate of pay and the weekend hourly rate of pay?" (2 marks)
• 2018 NHT Exam 2 Question 3b — "Explain why this equation cannot be solved using the matrix inverse method." (rectangular 2 × 3 coefficient matrix)

1.5b Exam 1 forms

• 2008 Exam 1 Question 5 — "The determinant of [3 2; 6 x] is equal to 9. The value of x is"
• 2013 Exam 1 Question 4 — "2.8x + 0.7y = 10 / 1.4x + ky = 6 — The set of simultaneous linear equations above does not have a solution if k equals"
• 2015 Exam 1 Question 3 — "How many of these systems of simultaneous linear equations do not have a unique solution?"
• 2016 Exam 1 Question 3 — "These simultaneous linear equations have no unique solution when m is equal to"
• 2018 Exam 1 Question 1 — "Which one of the following matrices has a determinant of zero?"
• 2021 Exam 1 Question 3 — "ax + 4y = 10 / 18x + by = 6. The set of simultaneous linear equations above does not have a unique solution when"
• 2022 NHT Exam 1 Question 5 — "…does not have a unique solution if q has the value"
• 2022 Exam 1 Question 5 — "Which one of the following matrix products could have an inverse?" (tests that only a square product is invertible)
• 2025 Exam 1 Question 27 — "Consider the matrix E where E = [m −9; 4 n]. For the inverse of E to exist, the values of m and n, respectively, cannot be" (76 %). Report: "For the inverse of E to exist the determinant ≠ 0. det = mn − (−9 × 4) = mn + 36. The product mn must be negative therefore m and n must have different signs."
• 2023 Exam 1 Question 30 — a four-statement true/false item; the 2023 report lists the statements verbatim: "1. All square matrices have an inverse. 2. The inverse of a matrix could be the same as the transpose of that matrix. 3. If the determinant of a matrix is equal to zero, then the inverse does not exist. 4. It is possible to take the inverse of an identity matrix. Statements 2, 3 and 4 are true." Only 39 % correct.
• 2015 Exam 1 Question 7 / Sample Q31 — "Matrix P has inverse matrix P⁻¹. Matrix P is multiplied by the scalar w (w ≠ 0) to form matrix Q. Matrix Q⁻¹ is equal to" (answer (1/w)P⁻¹; report gives the full proof)
• 2014 Exam 1 Question 9 — "A and B are square matrices such that AB = BA = I, where I is an identity matrix. Which one of the following statements is not true?"
• 2019 NHT Exam 1 Question 6 — "Matrix W is a 3 × 2 matrix. Matrix Q is a matrix such that Q × W = W. Matrix Q could be" (identity)
• 2024 NHT Exam 1 Question 27 — "Matrix V is an n × n matrix with a determinant equal to 1. The product of V × V⁻¹ will result in"
• 2025 NHT Exam 1 Question 26 — report: "The product of a matrix by its inverse will result in an identity matrix."
• 2026 NHT Exam 1 Question 27 — "T is a matrix such that FT is an identity matrix. Which one of the following is T?"
• 2012 Exam 2 Question 2b — inverse used as a decoder: "Show how the original matrix A can be regenerated from matrix C." Report: "A = B⁻¹C. Other equivalent forms of this answer were accepted."

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1.6 Solving simultaneous equations with matrices (including setting up from a word problem)

Marks: Exam 2 typically 1 mark for the set-up template + 1–2 marks for the solution. Exam 1: 1 mark. Presence: both exams, near-annual 2006–2023 NHT. Rare in the 2023+ core papers (2022 was the last main-paper Exam 2 appearance; 2023 NHT Q3d and 2018 NHT Q3 are the other late examples).

1.6a Exam 2 — complete the matrix template from the word problem

• 2006 Exam 2 Question 3a — "a. Write the equations in matrix form using the following template. _ = _" Report warning: "Many students did not deal well with the missing x term in the third equation. A common error was to put the zero in the wrong position on the third line or to simply leave a gap. Neither of these two options was awarded a mark."
• 2009 Exam 2 Question 2a — "a. Use the information in Table 1 to complete the following matrix equation by inserting the missing values in the shaded boxes."
• 2010 Exam 2 Question 3a — "These equations can be written equivalently in matrix form. / Complete the missing information below."
• 2022 Exam 2 Question 2a — "a. Write the equations in matrix form by completing the following template. [20 40; 30 50] _ = _" (86 % correct)
• 2018 NHT Exam 2 Question 3ci — "Complete the matrix equation below by filling in the missing elements." (the inverse matrix, partially given)
• 2007 Exam 2 Question 1c — 4 × 4 solve: "Solve the matrix equation to find the values b, m, p and h." Report: "This question required accurate data entry into the calculator and then correct multiplication of the inverse of the square matrix by the column matrix. Errors in typing the matrix elements into the calculator seemed common. There was little room for method marks here, with two marks awarded for all four correct answers."

1.6b Exam 2 — extract the one value asked for

The recurring full-mark condition: give the number, not the column matrix.

• 2009 Exam 2 Question 2b — "b. Use the matrix equation to find the cost of a teacher ticket to the school function." Report: "It was not sufficient for students to simply provide the column matrix as the solution without extracting and writing the cost of the teacher's ticket as required by the question."
• 2017 Exam 2 Question 1ci — "i. What is the cost of one sandwich?" 56 %. Report: "Many students were unable to solve the matrix equation. Even those who did so correctly usually showed no working."
• 2022 Exam 2 Question 2c — "Complete the matrix below, showing the daily cost of hiring one set of skis and one snowboard. [x; y] = ____"
• 2018 NHT Exam 2 Question 3cii — "Determine the cost, in dollars, of one tonne of Phate (P)." → $1500
• 2010 Exam 2 Question 3c — "c. How many rebounds is Oscar predicted to have in his next game?"

1.6c Exam 1 — identify the correct matrix form

• 2007 Exam 1 Question 4 — "This system of equations can be written in matrix form as"
• 2009 Exam 1 Question 4 — "The matrix equation … can be used to solve the system of simultaneous linear equations"
• 2010 Exam 1 Question 5 — "A system of three simultaneous linear equations is written in matrix form as follows. … One of the three linear equations is"
• 2014 Exam 1 Question 2 — "The system of three simultaneous linear equations above can be written in matrix form as"
• 2017 Exam 1 Question 3 — "Which one of the following matrix equations has a unique solution?"
• 2019 Exam 1 Question 5 — "Which one of the following systems of simultaneous linear equations best represents the matrix equation above?"
• 2021 NHT Exam 1 Question 3 — "Which one of the following matrix equations has the same unique solution for x and y?"
• 2022 Exam 1 Question 6 — "The solution to these simultaneous equations can be found by calculating" (only 43 % chose the correct option; 31 % chose a distractor)
• Word-problem set-ups solved by CAS: 2010 Q3, 2013 Q6, 2015 Q6, 2018 Q5 (4 × 4), 2019 NHT Q4, 2017 NHT Q5 ("How many of the matrix equations above could Yvette solve …")

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1.7 Binary / communication matrices

What it looks like. A 4×4–6×6 matrix of 0s and 1s with a sender / receiver (or from / to) labelling, plus two worked "In this matrix:" bullets. Questions ask who can reach whom, the sequence of links, and what a two-step matrix M² element means.

Marks: 1 mark per part, 2–3 marks total. Presence: both exams, essentially every year.

1.7a Read the matrix

• 2016 Exam 2 Question 2a–b — "The '1' in row E, column D of matrix G indicates that Elka (sender) can send a direct message to Dana (receiver). The '0' in row E, column C of matrix G indicates that Elka cannot send a direct message to Cheng. / a. To whom can Dana send a direct message? / b. Cheng needs to send a message to Elka, but cannot do this directly. Write down the names of the employees who can send the message from Cheng directly to Elka."
• 2021 Exam 2 Question 2a–b — "a. Which two staff members can send information directly to each other? / b. Elena needs to send documents to Chai. What is the sequence of communication links that will successfully get the information from Elena to Chai?" Report answers: "Brie and Dex"; "Elena – Dex – Brie – Chai."
• 2023 Exam 2 Question 10a — "a. A customer wants to make a complaint to a director. / What is the shortest communication sequence that will successfully get this complaint to a director?"
• 2024 NHT Exam 2 Question 10a–c — "a. Which barn(s) can directly communicate to barn L? / b. State two communication paths for barn J to communicate to barn K. / c. The farmer plans to install one new direct communication link in one barn … In which barn should the new direct communication link be installed to ensure that barn I has either a one-step or a two-step communication link with every other barn?"
• 2022 NHT Exam 2 Question 2a — "a. Which location is directly connected by paths to more locations than any other?"
• 2012 Exam 2 Question 1a–b — "a. Complete the following sentence. On this airline, you can fly directly from Berga to _ and _. / b. List the route that you must follow to fly from Anvil to Cantor."
• 2013 Exam 2 Question 1c — "Complete matrix N below by filling in the missing elements in row 1 and column 1." (edges removed ⇒ symmetric matrix must be updated in both row and column)
• 2011 Exam 2 Question 1a–b (decoded) — "i. what does the '1' in column B row L of matrix E indicate? / ii. what does the row of zeros in matrix E indicate?" Report: "Birds eat lizards." / "No birds, lizards or insects eat birds."

1.7b Two-step communication, M² and M + M²

• 2021 Exam 2 Question 2c — "Matrix M² below is the square of matrix M and shows the number of two-step communication links between each pair of staff members. / Only one pair of individuals has two different two-step communication links. / List each two-step communication link for this pair." 21 % correct. Report: "Alex – Brie – Dex and Alex – Elena – Dex. This question was not answered well. Most students appeared not to interpret this question correctly."
• 2023 Exam 2 Question 10b — "b. Matrix H below shows the number of two-step communication links between each group. Sixteen elements in this matrix are missing. / i. Complete matrix H above by filling in the missing elements. / ii. What information do elements g₂₁ and h₂₁ provide about the communication between the circus employees?"
• 2022 NHT Exam 2 Question 2b–c — "b. When matrix W² is calculated, the element in row L, column R is equal to 2. Write down the paths represented by this element. / c. In matrix W + W², the values in row F, column R and row R, column F are both zero. Explain the significance of this." Report: "There are no one-step or two-step paths linking fitness and residential."
• 2019 NHT Exam 2 Question 2 — same structure ("When matrix W² is calculated, the element in row L, column R is equal to 2 … In matrix W + W², the values in row F, column R and row R, column F are both zero.")
• Exam 1: 2009 Q5, 2013 Q5, 2017 Q2, 2017 NHT Q3, 2019 Q7 ("In how many ways can Y get a message to W by sending it directly to one other person?"), 2020 Q5, 2022 Q4, 2023 NHT Q5, 2025 Q28, Sample Q25.

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1.8 Dominance matrices and ranking (including two-step dominance)

What it looks like. A round-robin result matrix; rank by the sum of one-step and two-step dominances (i.e. row sums of D + D²). From 2016 this is a recurring Exam 1 type; it entered Exam 2 in 2021.

Marks: 1 mark (Exam 1); 2 marks (Exam 2 2021). Presence: predominantly Exam 1; only one Exam 2 appearance in the corpus (2021 Q4).

• 2021 Exam 2 Question 4 (2 marks) — "Five staff members in Elena's office played a round-robin video game tournament, where each employee played each of the other employees once. In each game there was a winner and a loser. A table of their one-step and two-step dominances was prepared to summarise the results. … A '1' in this matrix shows that the player named in that row defeated the player named in that column. A '0' in this matrix shows that the player named in that row lost to the player named in that column. / Use all of the information provided to complete the results matrix." Report: "Only some students were successful in filling in the entire matrix" — 70 % scored 0.
• 2016 Exam 1 Question 8 — "In this tournament, each team was given a ranking that was determined by calculating the sum of its one-step and two-step dominances. … Team A would have been ranked number one (1) if the winner of one match had lost instead. That match was between teams". Report method: "A routine way of solving this problem was to construct a matrix whose row sums gave the total dominance (one- and two-step) scores for each player."
• 2017 Exam 1 Question 5 / Sample Exam 1 Question 27 — "Four teams, A, B, C and D, competed in a round-robin competition where each team played each of the other teams once. There were no draws. … In this matrix, the values of f, g and h are"
• 2017 NHT Exam 1 Question 7 — two-step reasoning: "If Amanda has 2 two-step dominances over Ben, she must have one via Carlos and one via Darius." (report solution)
• 2018 Exam 1 Question 3 — "There is an error in the matrix. The winner of one of the matches has been incorrectly recorded as a '0'. This match was between"
• 2020 Exam 1 Question 9 — "The winner of this darts tournament is the competitor with the highest sum of their one-step and two-step dominances. Which player, by winning their remaining match, will ensure that they are ranked first…?"
• 2022 NHT Exam 1 Question 6 — "Which one of the following tables shows the number of one-step and two-step dominances accumulated by each player in the tournament?"
• 2023 Exam 1 Question 28 — "A dominance matrix can display the results of each game, where a '1' in the matrix shows that the team named in that row defeated the team named in that column."
• 2024 GM Exam 1 Question 31 — a long four-case ranking argument; only 53 % correct (full reasoning in 2025-03_2024generalmaths1-report.txt).
• 2025 Exam 1 Question 32 — "The winner of the tournament was determined by T = D + D² where D and D² are, respectively, the one-step and two-step dominance matrices. Some of the individual match results were not recorded. … Which one of the following is matrix T?" (63 %)
• 2026 NHT Exam 1 Question 32 — "In order to break the tie between teams H and M, a new score, equal to the sum of the one-step and two-step dominance totals, is calculated for each team."

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1.9 Permutation matrices

What it looks like. A 0/1 matrix with exactly one 1 per row and per column, applied to a column matrix of letters, names, timetable slots or shifts. Pᵏ cycling, Pᵀ reversing, and "P⁴ is an identity matrix" all appear.

Marks: 1 mark. Presence: Exam 1 nearly every year since 2015; Exam 2 from 2023 (it is an explicit 2023+ study-design item).

• 2025 Exam 2 Question 14 (3 marks) — the fullest treatment. "Matrix P, shown below, is a permutation matrix used to determine the timetabled order of activities from one day to the next. … a. State the activities that are always held at the same time on each day of the program. (65 %) / b. Determine the timetabled order of the seven activities on day three of the program. (27 %) / c. P⁴ is an identity matrix. Explain what this means for the timetabled order of the activities over the 40-day holiday program. (11 %)" Assessment guide answer for (c): "Order of the activities rotates on a 4-day cycle OR There are 10 cycles completed | OR equivalent — Ongoing, repeating, 4-day cycles." Report: "Only a small proportion of students showed an understanding of the effect of the identity matrix." For (b) the guide adds: "Accept D, C, G, L, M, S, R / Accept column matrix / MUST be this order."
• 2023 GM Sample Exam 2 Question 13 — "a. When matrix N is multiplied by the permutation matrix, P, the cleaning roster will change from Week 1 to Week 2. Write down permutation matrix P in the space provided. / b. The transpose of the permutation matrix, Pᵀ, was used to create a new roster for March. … Complete the cleaning roster for weeks 2, 3 and 4 in March."
• 2023 Exam 2 Question 9c–d — a permutation/cycle matrix in all but name: "c. The circus started in town I. What is the order in which the circus will visit the five towns? / d. … Complete the three columns in the following matrix, showing the new movement between the six locations."
• 2023 NHT Exam 2 Question 3 — "The following permutation matrix, P, is used to rotate the morning shifts, afternoon shifts and evening shifts. … a. On which day(s) will Bill's shifts never change? / b. On which day(s) will Bill be working a morning shift in Week 2? / c. … Write the permutation matrix that is multiplied by Matrix N."
• 2007 Exam 1 Question 8 — the ancestral version, presented as a transition matrix: "He then determined the answers to the remaining questions by following the transition matrix … The answers that he gave to the six test questions, starting with D, were". Report gives the diagram-based solution method.
• 2015 Exam 1 Question 5 — "The transition matrix that, starting tomorrow, ensures Wendy buys flowers in alphabetical order (C, D, I, R, T) is"
• 2016 Exam 1 Question 2 — "The matrix product [permutation] × [L; E; A; P; S] is equal to"
• 2017 Exam 1 Question 4 — "A permutation matrix, P, can be used to change [F E R S] into [A F R E S]. Matrix P is"
• 2018 Exam 1 Question 4 — "This matrix product results in the entire first and third rows of matrix W being swapped. The permutation matrix P is"
• 2018 NHT Exam 1 Question 3 — password scrambling: "Which one of the following is the scrambled password?"
• 2019 Exam 1 Question 4 — a 10-answer chain through T
• 2020 Exam 1 Question 3 — powers: "If Pⁿ × W = [A; S; T; O; R], the value of n could be"
• 2021 Exam 1 Question 4, 2021 NHT Q4, 2022 Q7 ("the element m₃₁ moves to element"), 2022 NHT Q4, 2023 Q26, 2026 NHT Q29 ("A permutation matrix, P, can be used to reorder [L I S T E N] to [S I L E N T] with one calculation.")
• 2024 NHT Exam 1 Question 30 — "A permutation matrix, P, is multiplied by matrix R to form the product matrix Q = PR. If Q is equal to R, how many different permutation matrices could have been used?" Report: "If Q = R, this implies that multiplying by the permutation matrix P does not change the position order of the elements T, A, L, L, Y in matrix R."
• 2019 Exam 2 Question 2d — permutation matrix under a different name: "Matrix V is similar to matrix T but has the first two rows of matrix T interchanged. The matrix product that will generate matrix V from matrix T is V = M × T where matrix M is a binary matrix. / Write down matrix M." 21 % correct. Report: "Many responses were incorrect. Some gave the identity matrix; others gave various assortments of 1s and 0s."

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1.10 Transition matrices and Markov chains

This is the largest single block in the topic: in Exam 2 it typically takes 6–9 of the 12–15 marks.

1.10a Reading a transition diagram ↔ constructing the matrix

Marks: 1–2 marks. Presence: both exams, essentially every year.

Two directions:

(i) Diagram → matrix. - 2006 Exam 2 Question 2a — "a. Enter this information into transition matrix T as indicated below (express percentages as proportions, for example write 76 % as 0.76)." (2 marks) Report: "Some students wrote percentages despite being asked for proportions. Some simply copied the figures in order, as printed into the three columns. The percentage, 8 %, was sometimes written as a proportion of 0.8." - 2023 GM Sample Exam 2 Question 10b — "b. Write the transition matrix, T, that represents the transition diagram above." - 2023 Exam 2 Question 11a — "a. Write down matrix T, the transition matrix, for this recurrence relation. ☐ this show / W L … W next show L" - Exam 1: 2009 Q8, 2012 Q5, 2016 Q6, 2020 Q4, 2021 Q2, 2022 Q3, 2022 NHT Q3.

(ii) Matrix → diagram (complete the diagram). - 2009 Exam 2 Question 3b — "i. Complete the diagram above by writing the missing percentage in the shaded box." - 2013 Exam 2 Question 2aii (2 marks) — "ii. complete the transition diagram below, showing the relevant percentages." Report: "The 100 % cycle drawn at D (all that die this year will be dead next year) was a common omission. Students should not add edges marked 0 % that are in the opposite direction to the given edges. Similarly, loops with 0 % at E and B should not be included." - 2015 Exam 2 Question 3a (2 marks) — "An incomplete transition diagram for matrix T₂ is shown below. / Complete the transition diagram by adding the missing information." Report: "One mark was available for a transition from I to L, including the arrow and the 10 % label. Another mark was available for the loop at L, including the 100 % label. A number of students missed the loop entirely, while others incorrectly labelled it as 1 %. Some students incorrectly added a second 5 % directed transition from B to A. Others unnecessarily added a lot of 0 % transitions in the reverse directions for all shown transitions." - 2017 NHT Exam 2 Question 2a (2 marks) — "Complete the transition diagram above by adding all the missing information." Report: "One mark each was awarded for: arrowhead from C to A; arrowhead from C to B and labelled with 0.65." - 2022 NHT Exam 2 Question 3a — "Complete the transition diagram by writing the three missing percentages in the boxes provided." - 2023 NHT Exam 2 Question 2b — "Complete the transition diagram above by adding all the missing values, as percentages, on the edges." - 2024 NHT Exam 2 Question 12a — "Complete the transition diagram by adding the missing edges and the missing proportions." - 2024 GM Exam 2 Question 11ai — the biggest recent failure: 24 % correct. Report: "This question was not well answered. All edges required arrows otherwise it is no longer a directed graph." Assessment guide: "Accept percentage values / Accept additional edges with zero value."

1.10b Interpret a single element / a loop / a "1" in the matrix

Marks: 1 mark. A guaranteed low-scoring question. (See §2.3 for the required answer form.)

• 2010 Exam 2 Question 2a — "a. What information does the 20 % in the diagram above provide?" Report: "20 % of those doing heavy training one week will change to moderate training the next week. Some students seemed confused about what the given percentage referred to. A common incorrect answer was '20 % of players will change to moderate training the next week'. This would mean that 20 % of a fixed number of 300 will change, whereas it is only 20 % of a variable number (initially 90). Others suggested that the change would (only) occur in the first week rather than a regular transitional change."
• 2012 Exam 2 Question 3b — "b. Explain the meaning of the entry in the fourth row and fourth column of transition matrix T." Report: "Once a worker leaves the site, they never return (or equivalent statement). Most students were unable to explain the meaning of the 1.00 figure … The most common unacceptable answer was '100 % of those who leave this year will leave the next year', while others said that '100 % of the people who do job L this year and will again do job L next year'."
• 2015 Exam 2 Question 2a — "a. The element in the third row and third column of matrix T₁ is the number 1. / Explain what this tells you about the advanced-level students." → "All of the advanced-level students stay as advanced-level students."
• 2021 NHT Exam 2 Question 3a — "a. Explain the meaning of the value 0.29 in the transition matrix T." → "29 % of students obtaining a credit this year are expected to obtain a distinction next year."
• 2023 NHT Exam 2 Question 2a — "a. Interpret what the 0 in row 3, column 1 means." → "No driver upgrades from a light-rigid licence one year to a heavy-rigid licence the next year."
• 2023 GM Sample Exam 2 Question 10a — "a. Interpret the value on the loop at V."
• 2025 Exam 2 Question 13a — "a. What do the values on the leading diagonal in matrix K indicate?" (67 %) → "No child does the same activity in any two consecutive weeks." Guide: "No child does the same activity from one week to the next. | OR equivalent"
• 2025 NHT Exam 2 Question 13a–b — "No resorts are classified as Budget one year and then Luxury the next." / "There will not be any resorts classified as Budget." Guide adds "Or similar wording".
• 2016 Exam 2 Question 3ei — the negative-element variant: "The element in the fourth row of matrix B is −80. / Explain this number in the context of selecting customers for the studies in 2017 and 2018." 14 % correct. Report: "80 customers who have no bookings/no travel in that year will be removed from the study. Many students were unable to explain the −80 in matrix B. The answer needed to refer to the removal from the study of 80 people selected from the no travel group (N) in 2017 and in 2018. A common unacceptable response was, '80 people chose not to travel each year'."

1.10c "Show that" a single state-matrix element — the boxes question

A signature VCAA format: the target number is given; students fill boxes with the four transition proportions.

• 2016 Exam 2 Question 3b — "b. Write a calculation that shows that 478 customers were expected to choose air travel in 2015." 35 %. Report: "0.65 × 520 + 0.25 × 320 + 0.25 × 80 + 0.5 × 80 = 478 customers. A matrix product such as TS₀ does not show an understanding of how 478 is calculated."
• 2017 Exam 2 Question 2b — "306 junior students are expected to choose sport (S) in Term 2. / Complete the calculation below to show this. 300 × ☐ + 240 × ☐ + 210 × ☐ = 306" 69 %. Report: "Some students gave the answers as percentages. A few just gave 40, 60, 20."
• 2018 Exam 2 Question 3b — "b. Show that the length of highway that is to be graded (G) in 2019 is 460 km by writing the appropriate numbers in the boxes below. ☐ × 700 + ☐ × 400 + ☐ × 200 + ☐ × 1400 = 460" 77 %. Report: "Some students listed the numbers in the G column rather than the row."
• 2019 NHT Exam 2 Question 2a — "a. Complete the calculation below to show that 835 Gillen residents are not expected to watch television (NoTV) at 10.00 am that day. ☐ × 100 + ☐ × 400 + ☐ × 100 + ☐ × 1400 = 835"
• 2023 GM Sample Exam 2 Question 11a — "a. Complete the following multiplication for the row that determines the number of adult fish predicted to be in the population after one year. ☐ × 10000 + ☐ × 1000 + ☐ × 800 + ☐ × 0"
• 2021 NHT Exam 2 Question 3b — "b. Show that the value of k is 0.62" (column sums to 1)
• 2014 Exam 2 Question 1e — the non-transition analogue: "Using appropriate elements from the matrix product V × P, write a calculation to show that the value of w is 1415." Report: "1360 × 0.45 + 1460 × 0.55 = 1415. Some students copied the full given matrix multiplication and replaced the w with 1415. This was not acceptable."
• 2021 Exam 2 Question 3d — "d. Write a calculation that shows that Ohana olive oil is the brand bought by 50 % of these shoppers in the long run." 22 %. Report: "Many students found a long-term matrix but were unsure what to do with it to answer the question."

1.10d Backward-percentage questions ("of those at X next period, what % were at Y this period?")

The single most consistently failed Exam 2 matrices type. Success rates: 2016 — 17 %; 2017 — 30 %; 2018 — 17 %; 2019 — 26 %; 2021 — 17 %; 2022 — 25 %; 2024 NHT — asked twice.

The structure is always: numerator = one transition contribution; denominator = the whole new-state total, not the old-state total, and not a transition-matrix element read straight off.

• 2016 Exam 2 Question 3d — "Consider the customers who were expected to choose air travel in 2015. / What percentage of these customers had also chosen air travel in 2014? / Round your answer to the nearest whole number." Report: "478 customers were expected to choose air travel in 2015. 65 % of 520 who chose air travel in 2014 also chose air travel in 2015 i.e. 338 customers. Required percentage = 338/478 × 100 = 70.71… = 71 %. Common incorrect answers were 65 % and 94 %."
• 2017 Exam 2 Question 3c (2 marks) — "Of the senior students expected to choose investigation (I) in Term 3, what percentage chose service (S) in Term 2?" 86 % scored zero. Report: "200 expected in service in Term 2 / 30 % of 200 = 60 from service to investigation in Term 3 / 240 expected in investigation in Term 3 / 60/240 = 25 %. Some students were able to find either the 240 doing investigation or the 60 transitioning … but were unable to link these together."
• 2018 Exam 2 Question 3d — "In 2020, 1536 km of highway is expected to require no maintenance (N). / Of these kilometres, what percentage is expected to have had no maintenance (N) in 2019? / Round your answer to one decimal place." 17 %. Report: "48.5 %. Most students giving 97 %, which did not take account of the 0.5 change."
• 2019 Exam 2 Question 2c — "Of the 300 visitors expected at Ground World (G) at 11 am, what percentage was at either Air World (A) or Food World (F) at 10 am?" 26 %. Report: "(0.1 × 600 + 0.2 × 600)/300 = 0.6."
• 2021 Exam 2 Question 3c — "Consider the shoppers who were expected to buy Carmani olive oil in August 2021. / What percentage of these shoppers also bought Carmani olive oil in July 2021? / Round your answer to the nearest percentage." 17 %. Report: "89 %. 85 % taken directly from the transition matrix was a very common incorrect answer."
• 2022 Exam 2 Question 3b — "Consider the skiers who are expected to choose the advanced ski run on 17 July. / What percentage of these skiers also chose the advanced ski run on 16 July? / Round your answer to the nearest whole number." 25 %. Report: "63 %. Repeat skiers = 0.8 × 80 = 64. Total advanced skiers = 0.2 × 190 + 0.8 × 80 = 38 + 64 = 102."
• 2024 NHT Exam 2 Question 13b–c — "After two days, 8962 chickens are expected to be in North field. / How many of these 8962 chickens are expected to have been in North field initially?" and "Of the chickens expected to be in East field after three days, what is the percentage of chickens that were also expected to be in East field after both one day and two days?"
• 2023 GM Exam 1 Question 32 — the Exam 1 form, 18 % correct. Report: "From S₂, 127 students are expected to spend their lunch break at the oval. Of these students, 0.3 × 140 = 42 were at the oval on Tuesday. A common error in this question was to use the 0.3 (t₃₃) from the transition diagram, which gives the expected proportion of students who stay at the oval from one day to the next."
• 2019 NHT Exam 1 Question 7 — "Of the cows expected to be at Q in week 24 of 2019, the percentage of these cows at R in week 23 of 2019 is closest to"

1.10e Steady state / long run

Marks: 1 mark (usually), 2 marks when a "show" is demanded. Presence: both exams, essentially every year.

• 2006 Exam 2 Question 2d (2 marks) — the classic "show" form: "d. Show by calculating at least two appropriate state matrices that, in the long term, the number of customers expected at each centre each week is given by the matrix K = [194983; 150513; 254504]." Report: "Any two products Tⁿ K₀ where n ≥ 38. For example: K₃₈ = T³⁸·K₀ = … and K₃₉ = T³⁹·K₀ = … and this is the same as K₃₈."
• 2010 Exam 2 Question 2d — "d. Show that, after seven weeks, the number of players (correct to the nearest whole number) who are involved in each type of training will not change." Report: "In order to show a steady state had occurred 'after 7 weeks', it was necessary to demonstrate that a steady state had been achieved for two consecutive weeks. … Many students found S₇ and then, for example S₁₀₀, and incorrectly concluded that since S₇ = S₁₀₀ this fully justified the assertion that a steady state existed at S₇. However, such a pair of non-consecutive calculations did not eliminate the possibility that S₇ may not equal S₈."
• 2007 Exam 2 Question 2cv — "v. Write down the exact steady-state matrix for this population." Report: "Some students apparently did not raise their transition matrix to a high enough power to find this steady state matrix. Others did not interpret a calculator answer such as 5.8460065E−30 as representing zero or 699.999828 as representing 700 in this practical context. Such unrounded answers were not accepted."
• 2008 Exam 2 Question 2d — "d. In the long term, how many History students are predicted to attend lectures?" Report: "When two different calculations involving Sₙ = Tⁿ⁻¹ · S₁ produce the same result, the long-term state matrix has been produced. High powers for n, such as n = 50 and 51, could be used."
• 2012 Exam 2 Question 3civ — "iv. the total number of staff at the site in the longer term." → "No staff will remain at the site in the long-term."; "The most common incorrect answer was 350."
• 2017 Exam 2 Question 3d — "d. What is the maximum number of senior students expected in investigation (I) at any time during 2018?" 43 %. Report: "250. 239 was the most common incorrect response from the steady state matrix." (a "maximum", not a "long run" — see §3.6)
• 2018 Exam 2 Question 3e — "e. In the long term, what percentage of highway each year is expected to have no maintenance (N)? / Round your answer to one decimal place." 39 %. Report: "56.7 %. Some students left the answer as 1532.1 and did not convert to the required percentage."
• 2020 Exam 2 Question 3d — "d. In the long term, what is the expected weekly number of shoppers at Westmall? / Round your answer to the nearest whole number." 39 % → 218 884. "Some students gave 220 000, possibly from the graph."
• 2022 NHT Exam 2 Question 3e — "e. In the long run, how many of the 450 residents are expected to change their activity location from the cinema to the library each day?"
• 2017 NHT Exam 2 Question 2e — "e. In the long term, how many members would be expected to buy B class seats for a concert?" → 100
• 2026 NHT Exam 2 Question 12b — "b. In the long term, how many sugar gliders in total are expected to be in the wildlife park?" Guide: "1330 + 590 + 280 = 2200" (constant total ⇒ no iteration needed)
• 2024 GM Exam 2 Question 12b — absorbing state. 24 % correct. Report: "0. Students who recognised that the 1 in the transition matrix would result in an absorbing state performed well on this question." Guide: "TⁿS₂₀₂₃, (for very large n) gives 390 as having left".
• 2023 GM Sample Exam 2 Question 11biii — "iii. Which populations, E, B, A and D, remain the same in the long term?"
• Exam 1: 2006 Q9, 2007 Q6, 2008 Q8, 2010 Q7, 2013 Q3, 2014 Q3, 2016 Q7, 2017 NHT Q6, 2018 Q8, 2018 NHT Q8, 2019 NHT Q8, 2023 Q27, 2023 NHT Q8, 2026 NHT Q31. The 2018 Exam 1 report gives the recommended technique verbatim: "The long-term expectation involves raising the transition matrix to a high power, for example, 50. The steady state matrix is independent of the initial state, so one suggested approach is to use any initial state matrix where the values added to 500."

1.10f Sₙ = Tⁿ × S₀ and off-by-one on the power

• 2008 Exam 2 Question 2b — "b. Write down a matrix equation for Sₙ in terms of T, n and S₁." → Sₙ = Tⁿ⁻¹ · S₁; report: "The initial state matrix defined as S₁ in the question was required here."
• 2014 Exam 2 Question 2ci — "Evaluate the matrix S₃ = T²S₁ and write it down in the space below. / Write the elements, correct to the nearest whole number."
• 2014 Exam 2 Question 2d — report: "Many students used S_June = S₆ = T⁶ · S₁ to find the answer of 5404, which was incorrect. The state matrix for January was given as S₁. Then S₂ = T·S₁, S₃ = T²·S₁, … , Sₙ = Tⁿ⁻¹·S₁. The power by which the transition matrix must be raised is one less than the number of the required state matrix."
• 2025 Exam 2 Question 12a — the inverse direction: "The state matrix S₂₀₂₄ … gives the number of children in each category at the start of 2024. / Find S₂₀₂₃." (53 %) — requires M⁻¹S₂₀₂₄.
• 2026 NHT Exam 2 Question 12a (2 marks) — "How many sugar gliders were expected to be in zone J at the start of 2023?" Guide: "M1 Correct state matrix (A₂₀₂₄ or A₂₀₂₃); A1 Correct answer"
• 2024 GM Exam 1 Question 32 — the same reverse-iteration idea with +B: report "S₃ = TS₂ + A, hence S₂ = T⁻¹(S₃ − A)", iterated three times. 33 % correct.
• Exam 1: 2009 Q9 ("If S₃ = T S₂, then S₂ equals"), 2014 Q7, 2015 Q9 (two-phase: "S₈ = T₂³(T₁⁴S₁)"), 2019 Q6, 2020 Q7, 2021 Q8, 2022 NHT Q7, 2026 NHT Q30.

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1.11 Matrix recurrence relations: Sₙ₊₁ = T Sₙ and Sₙ₊₁ = T Sₙ + B

Marks: 1 mark for writing/completing; 1–2 marks for iterating. Presence: both exams; the + B form is examined every single year in Exam 2 since 2007 and is explicitly on the 2023+ formula sheet.

1.11a Plain Sₙ₊₁ = T Sₙ

• 2012 Exam 2 Question 3c — "c. Use the rule Sₙ₊₁ = T Sₙ to find i. S₂₀₁₂ …"
• 2017 Exam 2 Question 3 — "For the given matrix S₁, a matrix rule that can be used to predict the number of senior students in each elective activity in Terms 2, 3 and 4 is S₁ = […], Sₙ₊₁ = TSₙ / b. Complete S₂, the state matrix for Term 2, below." 73 %. Report: "Students who answered incorrectly tended to give the state matrix for Term 3."
• 2018 Exam 2 Question 3c — "The state matrix describing the highway maintenance schedule for the nth year after 2018 is given by Sₙ₊₁ = T Sₙ / c. Complete the state matrix, S₁, below…"
• 2019 Exam 2 Question 2 — "The number of visitors expected at each location n hours after 10 am on Saturday can be determined by the matrix recurrence relation below. S₀ = […], Sₙ₊₁ = T × Sₙ / b. Complete the state matrix, S₁, below…"
• 2021 Exam 2 Question 3b — "b. Using the rule Sₙ₊₁ = T × Sₙ, complete the matrix S₁ below."
• 2022 NHT Exam 2 Question 3b — "Complete matrix S₁ to show the number of residents participating in activities at each location on 2 January."
• 2025 Exam 2 Question 12a — "The number of children expected to be in each of the four categories, from one year to the next, can be calculated using the matrix recurrence relation Sₙ₊₁ = MSₙ where Sₙ represents the expected number of children in each of the four categories at the start of year n."

1.11b Sₙ₊₁ = T Sₙ + B (and the − B / + 500 M Sₙ variants)

This is where the marks are, and where most of the 0 % bands sit.

• 2007 Exam 2 Question 2d — the earliest: "S₁ = T S₀ + B S₀ … This pattern continues. The population matrix after n weeks, Sₙ, is given by Sₙ = T Sₙ₋₁ + B Sₙ₋₁"
• 2008 Exam 2 Question 3 — both + B and − D: "O₂₀₀₉ = A S₂₀₀₈ + B" then "Oₙ₊₁ = C Oₙ − D". Report: "The added column matrix at each year is applied after each transition matrix is applied. Consequently, simply squaring the transition matrix is not appropriate in this question."
• 2009 Exam 2 Question 4 — Lₙ₊₁ = T Lₙ − [5; 7]. Report gives the explicit non-identity: "L₃ ≠ T² · L₁ − [5;7]", and the interpretation mark: "The subtracted column matrix [5;7] indicates that, each week, another 5 + 7 = 12 students will no longer turn up to any rehearsal."
• 2012 Exam 2 Question 3d (2 marks) — "Sₙ₊₁ = T Sₙ + A … d. Find the expected number of operators at the site at the beginning of 2013." Report: "Many students instead found S₂₀₁₃ = T · S₂₀₁₁ + A rather than S₂₀₁₂ = T S₂₀₁₁ + A and therefore S₂₀₁₃ = T S₂₀₁₂ + A."
• 2013 Exam 2 Question 2e — Sₙ = T Sₙ₋₁ + 500 M Sₙ₋₁. Report: "Many students instead found S₂ = T · S₀ + 500 M S₀ rather than using S₁ to determine S₂. Of those who did a correct calculation, some left the matrix S₂ as their answer. The number of eggs had to be extracted from matrix S₂ for full marks."
• 2015 Exam 2 Question 3e (2 marks) — Rₙ₊₁ = T₂ × Rₙ + V. Report: "Of those who attempted this question, many inappropriately used R₃ = (T₂)³R₀ + V, whereas R₃ = TR₂ + V needed to be used instead. Some students found and interpreted R₄ instead of R₃."
• 2016 Exam 2 Question 3eii (2 marks) — R₂₀₁₇ = TR₂₀₁₆ + B, R₂₀₁₈ = TR₂₀₁₇ + B. Report: "Many students found the correct matrix R₂₀₁₈ but did not extract the answer 190 customers who chose sea travel. Some students only found R₂₀₁₇, and some of these students simply called it R₂₀₁₈ instead. A method mark for this two-mark question may have been earned for a correct and labelled matrix R₂₀₁₇, even if the final answer was incorrect."
• 2018 Exam 2 Question 4 — "Mₙ₊₁ = T Mₙ + B / a. Write down the value of k in matrix B. (39 %) / b. How many kilometres of highway are expected to be graded (G) in the year 2022? (23 %)" Report on (a): "Many students did not recognise that the 2700 km of highway remained constant throughout." On (b): "Some wrote the correct matrix for 2022 only and did not extract the number 457 as their answer."
• 2019 Exam 2 Question 3b — R₁ = V × R₀ + B₁, then R₂ = VR₁ + B₂ where B is the thing to find: "where matrix B₁ shows the required movement of visitors at 11 am. / i. Determine the matrix B₁." 25 %. Report: "[−210; 0; 210; 0] — Must take 210 from A and add to G. This question was poorly done, with many students apparently unable to identify what was required." Part ii: 12 %; "A few responses gave the matrix B₂ rather than the state matrix R₂."
• 2020 Exam 2 Question 4 — reverse iteration: "Rₙ₊₁ = TRₙ + B … a. Determine the expected number of shoppers at Westmall in the third week after it is sold. (50 %) / b. … in the first week after it is sold. (20 %)" Report both parts: "A small number of students wrote the matrix without indicating which element represented the answer."
• 2021 Exam 2 Question 3e — find the unknown element of B from a later state: "k represents the extra number of shoppers expected to buy Ohana olive oil each month. / If R₂ = [3333; 2025; 3642], what is the value of k?" 15 %.
• 2023 Exam 2 Question 11 (2 marks) — writing both matrices from prose: "This situation can be modelled by the matrix recurrence relation S₀ = [180; 0], Sₙ₊₁ = TSₙ + B / a. Write down matrix T, the transition matrix, for this recurrence relation. / b. Write down matrix B for this recurrence relation to ensure that the circus always has 180 workers."
• 2023 NHT Exam 2 Question 2d — B partially unknown and must be deduced from prose: "the transport company will employ an additional three light rigid vehicle licensed drivers and five medium rigid vehicle licensed drivers each year. They also anticipate that two heavy rigid vehicle licensed drivers will leave the company each year. … Matrix B is incomplete. / How many light rigid vehicle licensed drivers is the company expected to have in January 2025? / Round your answer to the nearest whole number."
• 2025 Exam 2 Question 12b — "find the expected total number of children to be enrolled in the early learning centre at the start of 2026. Round your answer to the nearest whole number." 13 %. Report: "15.25 + 15.93 + 19.11 = 50. Many students answered 115 by including the children who had left the centre."
• 2026 NHT Exam 2 Question 12 — Aₙ₊₁ = TAₙ + B, run backwards two years.
• 2019 NHT Exam 2 Question 4 — "Wₘ₊₁ = TWₘ + V … a. Write down matrix V. (1 mark) / b. How many people in Gillen are expected to watch C₂ at 7.00 pm on this day? (2 marks)"
• 2021 NHT Exam 2 Question 4 (2 marks) — reverse: "The state matrix for the number of questions in the file at the end of the fourth month is Q₄ = […]. How many multiple-choice questions were in the file at the end of the second month?"
• 2018 NHT Exam 2 Question 4 (2 marks) — Q must be deduced from H₀ and H₁ before H₂ can be found: "Hₙ₊₁ = R Hₙ + Q where … Q is a matrix that contains additional fixed changes … Complete H₂, the state matrix for 2020." Report: "A mark was available for finding matrix Q = [300; 300; −600]."
• 2022 NHT Exam 2 Question 3f — the +B idea hidden in prose: "The number of family members attending the cinema each day remained constant. / How many people are expected to attend the cinema on 3 March?"
• Exam 1: 2013 Q8, 2017 Q7 (fish farm, sign interpretation of B), 2018 Q7, 2020 Q10, 2021 Q7, 2024 NHT Q32, 2024 GM Q32.

1.11c "Write down a recurrence relation, in terms of …"

The exact phrase-frame VCAA uses. Note: in the corpus this frame appears with matrix state matrices only in the 2017 NHT paper; every other instance is the Recursion and financial modelling topic. Teachers should know the frame because it is the one VCAA would reuse if it ever asked for a matrix recurrence in words.

• 2017 NHT Exam 2 Question 2 (matrices module) — "Write down a recurrence relation, in terms of Sₙ₊₁ and Sₙ, that can be used to determine …"
• Financial-modelling instances of the same frame (for wording calibration): 2018 NHT ("in terms of V₀, Vₙ₊₁ and Vₙ"), 2019 ("in terms of V₀, Vₙ₊₁ and Vₙ" and "in terms of B₀, Bₙ₊₁ and Bₙ"), 2020 ("in terms of S₀, Sₙ₊₁ and Sₙ"), 2021 ("Complete the recurrence relation, in terms of S₀, Sₙ₊₁ and Sₙ"), 2022 ("in terms of P₀, Pₙ₊₁ and Pₙ"), GM Sample 2 ("in terms of S₀, Sₙ₊₁ and Sₙ").

The stable frame is therefore:

"Write down a recurrence relation, in terms of S₀, Sₙ₊₁ and Sₙ, that can be used to determine …"

and the full-mark answer must contain both the initial state and the rule: S₀ = [ … ], Sₙ₊₁ = T Sₙ (or T Sₙ + B).

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1.12 Leslie matrices and population growth (2023+ only)

Not examinable before 2023. Introduced with the General Mathematics study design; on the formula sheet as S₀ = initial state, Sₙ₊₁ = L Sₙ. Now a guaranteed presence: at least one Exam 1 question and often a whole Exam 2 question.

Marks: Exam 1 1 mark; Exam 2 3–4 marks.

• 2026 NHT Exam 2 Question 11 (4 marks) — the fullest example. "a. Interpret each of the three elements in the first row of matrix L. (1 mark)" Guide answer: "Female(s) are expected to have 0 offspring in their first year of life, 2 offspring in their second year of life, 1 offspring in their third year of life. | Accept equivalent statements" "b. What percentage of female numbats in their first year of life are expected to survive to their third year of life?" Guide: "0.35 × 0.22 = 0.077 → 7.7 %" "c. Showing relevant calculations, determine the level of threat faced by the numbat population 10 years later, at the start of 2032. (2 marks)" Guide: "2022 total population = 112 + 35 + 12 = 159; 2032 total population = 31.4 + 11.7 + 3.1 = 46; Population reduction = 159 − 46 = 113; Percentage reduction = 113/159 × 100 ≈ 71 % → Endangered | A1 H1 — Calculation for total population after 10 years (≈46) OR 71 % and 'endangered'. Correct percentage calculation and interpretation."
• 2024 GM Exam 2 Question 11 (2 marks-plus) — Leslie with four age groups 0–1, 1–2, 2–3, 3–4. "11a.i. complete the transition (life-cycle) diagram" — 24 %; "11a.ii. complete the table of initial population and population after one year" — 61 % (guide table: initial 70 / 80 / 90 / 40 → after one year 124 / 28 / 56 / 45); "11b. determine when the population would first be half the initial value" — 39 %, guide "131 < 0.5 × 280 → 5 years".
• 2023 GM Sample Exam 2 Question 12 (3 marks) — "The Leslie matrix, L, that models the breeding patterns for this female marsupial population is as follows. … The same information is presented in the transition diagram below. / a. The following table shows the birth rate, for each of the four age groups, the average proportion of offspring born to each age group, and the survival rate, the average proportion of offspring to survive to the next age group. Some values in the table are missing. Use the Leslie matrix to complete the table below. / b. By calculating S₁, determine how many marsupials there are aged 0–3 years after one three-year time period. / c. What percentage of this female marsupial population are aged 9–12 years after the second three-year time period? Round your answer to one decimal place."
• 2023 Exam 1 Question 31 — "The Leslie matrix, L, below is used to model the female population distribution of this species of bird. … The element in the second row, first column states that on average 20 % of this population will"
• 2024 GM Exam 1 Question 30 — 78 % correct. Report: "The Leslie matrix contains the birth rate for each age group in the first row. The survival rate for each age group will be shown in the next rows. The survival rate is from one age group to the next age group."
• 2024 NHT Exam 1 Questions 25 & 26 — "The following life cycle transition diagram shows changes in a female population of mammals with three age groups (1, 2 and 3). … On average, what percentage of the female population from group 2 will survive to group 3?" and "The associated Leslie matrix, L, for the above transition diagram is"
• 2025 Exam 1 Question 29 — 84 % correct. "The age groups, birth rates and survival rates are presented in the life-cycle diagram below. … Which one of the following is a Leslie matrix that corresponds with this life-cycle diagram?" Report: "Check row 1 for correct birth rates, no loop at year 1 so the birth rate for year 1 = 0."
• 2025 NHT Exam 1 Question 27 — long-run behaviour of a Leslie model: "Determine the Leslie matrix. Set up a sample calculation L × Sₙ, where the initial population is, for example, 400. By cycle number 25 the population will be 359, and by cycle 50 it will be 334. Total population is expected to decrease in the long term."
• GM Sample Exam 1 Question 29 — "The following Leslie matrix, L, can be used to model how a population of female animals of three age groups changes over time. … Given S₀ = [100; 0; 0], how many of the following statements are true?" (statements include "When k = 2, the population will decrease in the long term." / "When k = 3, the population will increase in the long term.")

Pre-2023 ancestors (the same mathematics, not called "Leslie"): - 2011 Exam 2 Question 3 (decoded) — "In this equation B is the breeding matrix B = [0 2; 0.25 0.5] … W₁ = BW₀ … Wₙ = BWₙ₋₁ … ii. Use matrices to determine the expected total number of female ducks in the colony in the long term. Write your answer correct to the nearest whole number." - 2013 Exam 2 Question 2 and 2023 GM Sample Exam 2 Question 11 — the egg/baby/adult/dead life-cycle transition matrix with an absorbing "D" state; structurally a Leslie-with-death model. - 2007 Exam 2 Question 2 — eggs (E), juveniles (J), adults (A), dead (D).

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2. THE VCAA WORDING BANK

These are the exact recurring phrasings, with the years that prove the repeat, and the full-mark answer structure the examination reports prescribe.

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2.1 Order of a matrix

Stems (all 1 mark):

Full-mark answer form. rows × columns, in that order, written as m × n. - 2014 report: "Some students reversed the two numbers, writing 2 × 4, which was not accepted." - 2016 report: "A common incorrect answer was 1 × 4. An answer of (4, 1) was not accepted."

2.2 "The element in row i and column j of matrix M, mᵢⱼ, …"

Stem frames: - "The element in row i and column j of matrix M is mᵢⱼ." — 2014 Ex1 Q6, 2015 Ex1 Q1, 2015 Ex1 Q8, 2016 Ex1 Q5, 2017 Ex2 Q1b, 2017 Ex1 Q6, 2017 NHT Ex1 Q4, 2019 Ex1 Q3, 2020 Ex1 Q2, 2020 Ex1 Q6, 2020 Ex2 Q1cii, 2022 Ex1 Q1, 2023 Ex2 Q8a, 2024 NHT Ex1 Q28, 2025 Ex1 Q30, Sample Ex1 Q28. - Follow-ups: "What does the element m₁₂ indicate?" (2017 Ex2) · "What does the element q₂₃ represent?" (2020 Ex2) · "Which element shows the cost for one child ticket?" (2023 Ex2) · "Interpret the value of element c₃₁." (2022 NHT Ex2) · "Explain what the matrix element R₂₄ represents." (2008 Ex2) · "Explain what the matrix element r₃₂ represents." (2021 NHT Ex2) · "Explain what the element in row 3, column 2 of matrix Rᵀ represents." (2019 Ex2 Q1e) · "Element m₂₁ indicates that" (2023 Ex1 Q25) · "The elements are determined by the rule" (construct-from-rule form).

Full-mark answer structure (prescribed by reports). The answer must name all three of: 1. the quantity counted (students / shoppers / cars / rolls), 2. the row category, and 3. the column category — in context, not as coordinates.

Evidence: - 2017 report (Q1b): "The number of rolls sold in week 1, which was 24. … Some students simply gave the element 24." - 2020 report (Q1cii): "The number of shoppers (594) in the clothing area of Westmall (at 1.00 pm). Most students identified the key information of shoppers, clothing and Westmall. A small number of students misread the matrix to give merchandise shoppers in Grandmall – reading q₃₂ instead of q₂₃." - 2019 report (Q1e): "The answer is the number of cars parked in area C for two hours. Some students recognised area C and 2 hours but did not refer to the number of cars parked." - 2024 GM Exam 2 assessment-guide video: "Question 10a required an understanding of element notation. Many students were unable to demonstrate an understanding of the notation in the question." (54 % correct) - 2021 NHT report: "The number of Year 9 students awarded a credit in English." (all three components present)

2.3 "Explain what the element … represents in the context of this problem"

Applied to a transition matrix element, the full-mark answer must be a conditional rate sentence, not a statement about a fixed count:

"[p] % of those who [state A] this [period] will [state B] the next [period]."

Prescribed by the 2010 report (Q2a), which is the clearest statement VCAA ever made:

"20 % of those doing heavy training one week will change to moderate training the next week. … A common
incorrect answer was '20 % of players will change to moderate training the next week'. This would mean that
20 % of a fixed number of 300 will change, whereas it is only 20 % of a variable number (initially 90).
Others suggested that the change would (only) occur in the first week rather than a regular transitional
change."

For a 1 on the leading diagonal (absorbing state), the required form is a permanence sentence: - 2012 report (Q3b): "Once a worker leaves the site, they never return (or equivalent statement). … The most common unacceptable answer was '100 % of those who leave this year will leave the next year'." - 2015 report (Q2a): "All of the advanced-level students stay as advanced-level students." - 2025 report (Q13a), leading diagonal of zeros: "No child does the same activity in any two consecutive weeks." Guide: "No child does the same activity from one week to the next. | OR equivalent"

For a 0 element, the required form is a negation in context: - 2023 NHT report (Q2a): "No driver upgrades from a light-rigid licence one year to a heavy-rigid licence the next year." - 2025 NHT guide (Q13a): "No (0) resorts are classified as Budget one year and then Luxury the next. Or similar wording" - 2011 report (Q1aii): "No birds, lizards or insects eat birds." — with the three ways to lose the mark listed: "no lizards or insects eat birds (this omitted one of the zeros) / insects do not eat insects, birds or lizards (confused a column with a row) / none of these animals eat their own kind (confused a diagonal with a row)."

2.4 "Write down a matrix expression / calculation that can be used to determine …"

The classic product-with-ones / cost-matrix question. Five stable variants:

What exactly earns the mark:

1. A matrix expression, not a number. The answer must be a product of two matrices written in the correct order. 2015 report: "C × Q. Also accepted was the matrix calculation written as: [15 25 40] × [5 10 3 2; 15 30 9 6; 10 20 6 4]. Some students incorrectly gave Q × C."
2. The order of multiplication must be conformable. 2020 report: "Many students gave the correct matrix calculation. Some had the two matrices but in the incorrect order."
3. Row vs column matrix matters and the labelling may be required. 2008 report: "Despite a row matrix being required, many students wrote a column matrix instead. The question expected students to label the matrix as shown."
4. When asked to "write and evaluate", both are needed. 2006 report (Q2c): "The expression and the result were both required for full marks."
5. Result of a matrix product is a matrix — keep the brackets. 2016 report: "[6000]. The product of two matrices produces a matrix. The brackets needed to be included."; 2017 NHT report: "[21 200]. The product of two matrices produces a matrix. The brackets must be included."; 2011 report: "The correct answer shown is a 1 × 4 matrix with the four elements clearly separated by spaces … Some students wrote their four elements separated by commas or dots; however, this is not correct mathematical notation."
6. When the target number is given, a generic matrix product does not "show" it. 2016 report (Q3b): "0.65 × 520 + 0.25 × 320 + 0.25 × 80 + 0.5 × 80 = 478 customers. A matrix product such as TS₀ does not show an understanding of how 478 is calculated."; 2014 report (Q1e): "Some students copied the full given matrix multiplication and replaced the w with 1415. This was not acceptable."
7. Matrix brackets are required even for a hand-written data matrix. 2007 report (Q1a): "Matrix brackets were required."

2.5 "Write down the transition matrix T" (from a transition diagram)

Full-mark answer form. - Columns must be the "this period" state, rows the "next period" state, and the printed row/column labels must be respected. 2006 report: "Some simply copied the figures in order, as printed into the three columns." - Entries as proportions if the question says proportions. 2006 report: "Some students wrote percentages despite being asked for proportions. … The percentage, 8 %, was sometimes written as a proportion of 0.8." - Each column must sum to 1 (used as a check and as a question in its own right — 2017 Ex1 Q8 report: "Two of the unknown values in the matrix T could be found from the elementary concept of transition matrices – that is, the column values must add to one."; 2020 Ex1 Q10 report: "Since the columns of the transition matrix add to 1, then …")

And the reverse (completing a transition diagram): - All edges must carry arrowheads. 2024 GM Exam 2 report: "This question was not well answered. All edges required arrows otherwise it is no longer a directed graph." The assessment-guide video is blunter: "This response was not awarded the mark due to the absence of the arrows. The use of percentages rather than proportions was acceptable." - Loops count and must be labelled. 2013 report: "The 100 % cycle drawn at D … was a common omission."; 2015 report: "Another mark was available for the loop at L, including the 100 % label. A number of students missed the loop entirely, while others incorrectly labelled it as 1 %." - Do not add 0 % reverse edges. 2013 report: "Students should not add edges marked 0 % that are in the opposite direction to the given edges. Similarly, loops with 0 % at E and B should not be included."; 2015 report: "Others unnecessarily added a lot of 0 % transitions in the reverse directions." (Caveat: the 2024 assessment guide says "Accept additional edges with zero value" — so the 0 %-edge penalty is not universal; it is applied where the diagram was already complete without them.)

2.6 "Write down a matrix recurrence relation, in terms of S₀ and Sₙ₊₁, that can be used to determine …"

Exact frames in the corpus: - "Write down a recurrence relation, in terms of Sₙ₊₁ and Sₙ, that can be used to determine …" — 2017 NHT Exam 2 (Matrices module) - "Write down a recurrence relation, in terms of S₀, Sₙ₊₁ and Sₙ, that could be used to model …" — 2020 Exam 2, GM Sample Exam 2 - "Complete the recurrence relation, in terms of S₀, Sₙ₊₁ and Sₙ, that would model …" — 2021 Exam 2 - "Write down a matrix equation for Sₙ in terms of T, n and S₁." — 2008 Exam 2 Q2b - Presented-to-the-student forms: "The matrix recurrence relation shown below can be used to determine / to calculate / to predict …" — 2018 Ex2, 2019 Ex2, 2020 Ex2 (×2), 2021 NHT Ex2, 2022 NHT Ex2, 2023 Ex2, 2023 NHT Ex1, 2025 Ex2, 2026 NHT Ex2. - "Consider the matrix recurrence relation below." — 2017 Ex1 Q8, 2020 Ex1 Q10, Sample Ex1 Q32.

Full-mark answer structure. Both pieces, as on the formula sheet:

S₀ = [initial column matrix],   Sₙ₊₁ = T Sₙ        (or  Sₙ₊₁ = T Sₙ + B)

The 2008 report is explicit that the paper's own subscript convention must be honoured: "Sₙ = Tⁿ⁻¹ · S₁ — The initial state matrix defined as S₁ in the question was required here."

2.7 "In the long term / in the long run" and steady-state wording

Full-mark answer structure for a "show steady state" question. VCAA requires two consecutive state matrices that agree: - 2010 report: "In order to show a steady state had occurred 'after 7 weeks', it was necessary to demonstrate that a steady state had been achieved for two consecutive weeks. … such a pair of non-consecutive calculations did not eliminate the possibility that S₇ may not equal S₈." - 2006 report: "Any two products Tⁿ K₀ where n ≥ 38 … and this is the same as K₃₈." - 2008 report: "When two different calculations involving Sₙ = Tⁿ⁻¹ · S₁ produce the same result, the long-term state matrix has been produced. High powers for n, such as n = 50 and 51, could be used." - 2018 Exam 1 report gives the shortcut: "The long-term expectation involves raising the transition matrix to a high power, for example, 50. The steady state matrix is independent of the initial state, so one suggested approach is to use any initial state matrix where the values added to 500."

When the answer is a description rather than a number (2010 Ex2 Q4c/d), the report demands the limiting value be named:

"Attendance at Dinosaurs' games is expected to gradually increase to 3000, at which level it will remain
steady. A description of 'the way in which the number of people attending the Dinosaur's games would change'
was required. The correct answer had to acknowledge that a steady state of 3000 would occur after about 80
weeks. Many students simply calculated the predicted attendance after 80 weeks and gave this number as their
answer. Others merely said that 'the attendance was increasing.' However, neither of these responses was
accepted."

2.8 "Explain why matrix … does not have an inverse" / determinant wording

Note: VCAA has never used the literal string "does not have an inverse" as a question stem in this corpus. The actual family is:

Full-mark answer structure. - "The determinant is not equal to zero" — not "greater than zero". 2020 report: "The determinant is not equal to zero. Many students did not recognise that a non-zero determinant is required for an inverse matrix to exist. Some stated that the determinant had to be greater than 0." - For the interpret-the-determinant form, both halves are required: 2022 report: "A unique solution exists, as the determinant is not zero. … The response also needed both parts." - For not defined, name the matrices and the mismatching counts: 2026 NHT guide: "Must reference M & Q OR 1 and 3 for columns/rows."; 2006 report warns that "second number / first number" phrasing "did not indicate whether the student knew which numbers referred to rows and which to columns."

2.9 "Show that …" in matrix questions

Three distinct shapes, each with a different full-mark requirement:

1. Show that a determinant equals a value — 2022 Ex2 Q2bi: "Show that the determinant of the matrix [20 40; 30 50] is −200." Report: "20 × 50 − 40 × 30. This question was a 'show that' question, which required the calculation to be clearly demonstrated."
2. Show that a state-matrix element equals a value — the boxes format (§1.10c). The individual products must be visible; a matrix product is not enough (2016 report).
3. Show that a steady state exists — two consecutive state matrices (§2.7).
4. Show that an unknown parameter equals a value — 2021 NHT Ex2 Q3b "Show that the value of k is 0.62"; GM Sample Ex2 Q11bi "If R₁ = [500; 4000; 650; 7150], show that k = 500."
5. Showing relevant calculations — 2026 NHT Ex2 Q11c: "Showing relevant calculations, determine the level of threat faced by the numbat population 10 years later". Guide assigns A1 and H1 for calculation + interpretation separately.

The 2024 assessment-guide video adds a useful mark-scheme note about how much working "show" really needs:

"Sample response for Question 12d. The first one here, the student successfully showed each step towards the
solution. Underneath that, the second student simply gave the correct answer. Both of these samples were
awarded the mark." (i.e. for a 1-mark question a bare correct answer suffices; for a 2-mark question it
does not — see §2.10.)

2.10 Rounding and "write your answer to the nearest whole number"

Exact instruction wordings (Exam 2):

Assessment-guide marking annotations (2023+) now state the rounding rule as a marking condition: - 2025 GM Exam 2 guide: "| Whole number rounding applies" (Q12b, Q13bii) and "| One decimal place rounding applies" (Q13bi). - 2022 report uses the same language in prose: "Whole number rounding applied for this question." (Q3b, Q3c, Q4b).

The prescribed rounding discipline: - Round only at the final step. 2006 report: "Rounding off should only occur at the final answer stage in a question, not halfway through the calculation." 2009 report: "A number (for example, length) found during working out within a question should not be rounded off when used to determine the subsequent answer for that question." 2025 report (Q13bi): "Rounding should only be done at the final step. The expected number of children should not have been rounded to 10." (only 9 % scored this mark) - But a rounded answer carried from a previous part is not penalised. 2006 report: "in cases where the answer to a previous question must be used, the rounded version of the previous answer may be used without penalty. (Students could also use the unrounded version of the previous answer without penalty.)" Repeated verbatim 2007, 2009. - Rounding down instead of to nearest loses the mark. 2015 report (Q3c): "Some students rounded 42.55 to 42 despite the question instruction, 'Write your answer correct to the nearest whole number'." - Don't over-round a population count that is genuinely fractional. 2013 report (Q2bii): "On average, the state matrix predicts that there will be 331.25 adult trout after four years. Since this average is a predicted value, a decimal number of adult trout is valid. An answer of 331 adult trout was common and also accepted." - Calculator floating-point noise must be read as exact. 2007 report (Q2cv): "Others did not interpret a calculator answer such as 5.8460065E−30 as representing zero or 699.999828 as representing 700 in this practical context. Such unrounded answers were not accepted." Also listed in that year's Areas of weakness as "interpreting a calculator result of something such as 4.8E-34 as zero in context". - Convert to a percentage when asked. 2015 report (Q3b): "Many students did not convert 21 students into the required percentage."; 2018 report (Q3e): "Some students left the answer as 1532.1 and did not convert to the required percentage."

Working requirement for multi-mark questions (2025 report, verbatim):

"For questions worth more than one mark, students are advised to show working. An incorrect answer on its
own will not be awarded any marks. A method mark can, where appropriate, be awarded to students who have
shown the development of their answer."

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3. TRAPS AND EXAMINER COMPLAINTS

Ordered by how often and how expensively they recur.

3.1 Order of multiplication (AB vs BA)

• 2015 Exam 2 Q1ci — "Some students incorrectly gave Q × C." (correct: C × Q)
• 2020 Exam 2 Q1d — "Many students gave the correct matrix calculation. Some had the two matrices but in the incorrect order."
• 2006 Exam 2 Q1c — the whole point of the question; report: "Many answers illustrated a poor understanding of the need for the number of columns in matrix P to be the same as the number of rows in matrix Q."
• 2017 NHT Exam 1 Q8 report — a full worked taxonomy of which pairings are defined: "AR is not defined / CA is not defined / CR is not defined / RC can give a 1 × 1 matrix but then matrix A cannot be further included as a third matrix / AC can give a column matrix / RA can give a row matrix."
• 2022 Exam 1 Q8 report — distinguishes the five options by what each product means, not just whether it is defined: "Option A is correct. Option B finds the total time for the four different departments. Option C finds the time for each of Henry, Irvine or Jean to service the laptops separately from the desktops. Option D finds the time for each … at the four different departments. Option E finds the total time altogether."
• 2026 NHT Exam 2 Q10 — the paper now teaches the trap directly: (a) "Explain why MQ is not defined", (b) "R = QᵀM".

3.2 Row matrix vs column matrix; transposing rows and columns

• 2008 Exam 2 Q1ci — "Despite a row matrix being required, many students wrote a column matrix instead. The question expected students to label the matrix as shown."
• 2017 Exam 2 Q1cii — "Some were correct but many students gave a 3 × 1 matrix" (1 × 3 required). Only 28 % correct.
• 2018 Exam 2 Q1c — "Some students gave the correct numbers but in a column matrix."
• 2018 Exam 2 Q3b — "Some students listed the numbers in the G column rather than the row."
• 2025 Exam 2 Q11b — "Many students gave an answer that suggested that the rows had been summed and not the columns."
• 2020 Exam 2 Q1cii — "A small number of students misread the matrix to give merchandise shoppers in Grandmall – reading q₃₂ instead of q₂₃."
• 2011 Exam 2 Q1aii — "insects do not eat insects, birds or lizards (confused a column with a row) / none of these animals eat their own kind (confused a diagonal with a row)."
• 2014 Exam 2 Q1a / 2016 Exam 2 Q1a — order reversal: "Some students reversed the two numbers, writing 2 × 4, which was not accepted." / "A common incorrect answer was 1 × 4."
• 2020 Exam 2 Q2b — inverse of a 2 × 2: "Some students did not include the negative signs, and others did not correctly interchange the positions of the 5 and 7."
• 2022 NHT Exam 1 Q8 report — "The answer requires T, U and V to be in the rows. It is necessary to first transpose matrix N."

3.3 Not answering in context / giving the matrix instead of the answer

This is the most frequently repeated complaint in the whole corpus.

• 2007 areas of weakness — "giving only the entire matrix as the answer (Question 2dii.) rather than explicitly answering the question that required identification of an element of that matrix"; and in the body: "Simply stating the resulting 4 × 1 matrix did not answer this question."
• 2009 Exam 2 Q2b — "It was not sufficient for students to simply provide the column matrix as the solution without extracting and writing the cost of the teacher's ticket as required by the question."
• 2013 Exam 2 Q2eii — "some left the matrix S₂ as their answer. The number of eggs had to be extracted from matrix S₂ for full marks."
• 2014 Exam 2 Q2d — "Some students wrote the complete matrix as their answer. This did not demonstrate an understanding of the correct required element in the matrix."
• 2016 Exam 2 Q3eii — "Many students found the correct matrix R₂₀₁₈ but did not extract the answer 190 customers who chose sea travel."
• 2018 Exam 2 Q4b — "Some wrote the correct matrix for 2022 only and did not extract the number 457 as their answer."
• 2020 Exam 2 Q4a and Q4b — "A small number of students wrote the matrix without indicating which element represented the answer." (repeated for both parts)
• 2019 Exam 2 Q3bii — "A few responses gave the matrix B₂ rather than the state matrix R₂."
• 2010 Exam 2 Q1c — context specificity: "Explanations should have been applicable to the context of the question. Since there were different points available for different types of shots at goal, an answer of 'The total of all the goals shot by Oscar' was not accepted."
• 2007 Exam 2 Q1biii — "This question referred to the ingredients to make one single sandwich. Many students incorrectly referred to the 'total energy in these four foods' and did not specify it was for the particular quantity of 'these four foods' needed to make one sandwich."
• 2012 Exam 2 Q1d — "It refers specifically to the number of direct flights out of each of the four cities to another city in the network, not the inward flights or just 'connections'."
• 2011 Exam 2 Q2d — "A variety of incorrect answers suggested that many students may benefit from increased practice interpreting the result of a matrix product within a context. … Incorrect answers often included the number of insects killed or suggested that the answer referred to the number of birds, lizards and frogs that survived."
• 2019 Exam 2 Q1e — "Some students recognised area C and 2 hours but did not refer to the number of cars parked."
• 2014 Exam 2 Q2cii — "An incorrect answer given by some students was 'The number of voters expected to change their votes in March'." (correct: the number of preferences for each candidate in March)

3.4 Rounding intermediate steps

See §2.10 for the rules. The failures: - 2025 Exam 2 Q13bi (9 % correct) — "Rounding should only be done at the final step. The expected number of children should not have been rounded to 10." - 2015 Exam 2 Q3c — "Some students rounded 42.55 to 42 despite the question instruction." - 2017 Exam 2 Q2c — "Those who calculated the figures for Term 4 often ended up with a total of 749 due to rounding or did not give the final answer as a total." - 2007 Exam 2 Q2cv — floating-point noise not interpreted (5.8460065E−30 ≠ zero; 699.999828 ≠ 700). - 2006/2007/2009 general reports — the same paragraph three years running: "Rounding off should only occur at the final answer stage in a question, not halfway through the calculation."

3.5 The + B recurrence: wrong power, wrong base, applying B before T

• 2008 Exam 2 Q3b — "The added column matrix at each year is applied after each transition matrix is applied. Consequently, simply squaring the transition matrix is not appropriate in this question."
• 2009 Exam 2 Q4a — "A common error was for students to square the transition matrix first and then use the initial state matrix. Students should note that: L₃ ≠ T² · L₁ − [5;7]."
• 2012 Exam 2 Q3d — "Many students instead found S₂₀₁₃ = T · S₂₀₁₁ + A rather than S₂₀₁₂ = T S₂₀₁₁ + A and therefore S₂₀₁₃ = T S₂₀₁₂ + A." Plus a second warning: "Other students inappropriately used the matrix S₂₀₁₂ from Question 3ci … This matrix for S₂₀₁₂ cannot be applied here since it was determined from a different relation that did not involve matrix A."
• 2013 Exam 2 Q2eii — "Many students instead found S₂ = T · S₀ + 500 M S₀ rather than using S₁ to determine S₂."
• 2015 Exam 2 Q3e — "many inappropriately used R₃ = (T₂)³R₀ + V, whereas R₃ = TR₂ + V needed to be used instead. Some students found and interpreted R₄ instead of R₃."
• 2016 Exam 2 Q3eii — "Some students only found R₂₀₁₇, and some of these students simply called it R₂₀₁₈ instead."

3.6 Off-by-one in the power of T and in the state-matrix subscript

• 2008 Exam 2 Q2ai — "A common mistake resulted from students misreading the question. The state matrix S₁ was given and so a single application of the transition matrix to find S₂ was required. Many students renamed the state matrix S₁ as S₀ and then squared the transition matrix."
• 2010 Exam 2 Q2b — "Some students relied on a formulaic approach and squared the transition matrix to find S₂. Such students incorrectly applied n = 2 and used S₂ = T² · S₀. However, the initial state matrix was given as S₁ rather than S₀ and so S₂ = T · S₁."
• 2010 Exam 2 Q2c — "A common incorrect answer was 3, found by using T³ instead of T²."
• 2010 Exam 2 Q4b — "The attendance matrix for game 10 is A₁₀. A₁₀ = G⁹A₁ … A common error was to use the incorrect power for the matrix G."
• 2014 Exam 2 Q2d — the clearest statement of the rule: "The power by which the transition matrix must be raised is one less than the number of the required state matrix."
• 2014 Exam 2 Q3b — "The most common error was S_June = T₁⁵ · S₁ or S_June = T₁⁶ · S₁ rather than S_June = T₁ · S_May." (80 % scored zero)
• 2012 Exam 2 Q3cii — "The most common incorrect answer was 170, found from S₂₀₁₂ rather than from S₂₀₁₃."
• 2017 Exam 2 Q3b — "Students who answered incorrectly tended to give the state matrix for Term 3."
• 2015 Exam 1 Q9 report — the two-phase version: "In answering this question many students did not take into account that the transition matrix changed from T₁ to T₂ after week 5."

3.7 Reading a transition-matrix proportion instead of computing the backward percentage

• 2021 Exam 2 Q3c — "85 % taken directly from the transition matrix was a very common incorrect answer."
• 2016 Exam 2 Q3d — "Common incorrect answers were 65 % and 94 %." (65 % is the matrix element)
• 2023 GM Exam 1 Q32 — "A common error in this question was to use the 0.3 (t₃₃) from the transition diagram, which gives the expected proportion of students who stay at the oval from one day to the next."
• 2018 Exam 2 Q3d — "most students giving 97 %, which did not take account of the 0.5 change."
• 2014 Exam 2 Q3a — "Many students seemed unable to identify the sub-group who were expected to change their votes … The most common incorrect answer was 5 %, which represents the percentage of Mr Broad's total votes that reverted back to Mr Broad."
• 2017 Exam 2 Q3c — "Some students were able to find either the 240 doing investigation or the 60 transitioning from service to investigation but were unable to link these together."

3.8 Confusing "maximum", "first time", "long run" and "total"

• 2017 Exam 2 Q3d ("maximum number … at any time during 2018") — "239 was the most common incorrect response from the steady state matrix."
• 2022 Exam 2 Q3c ("maximum number of students expected to ski the intermediate ski run on any one day") — "219. The maximum occurred three days after the original date. Many students incorrectly assumed that the long run number was required."
• 2013 Exam 2 Q2cii ("largest number of adult trout … in any one year") — "It was not sufficient to assume that the number of adult trout was decreasing without checking the values given by S₂ and S₃. A common incorrect answer was 800, taken directly from S₀."
• 2012 Exam 2 Q3civ ("total number of staff at the site in the longer term") — "The most common incorrect answer was 350."
• 2012 Exam 2 Q3ciii ("the beginning of which year the number of operators first drops below 30") — "An answer of 'the 10th year' was not accepted, since actual years were used and required." Compare 2024 GM Exam 2 Q12d, where the assessment guide now says "| Accept fourth year or four years" alongside "2027" — the convention has softened but the safe answer is the calendar year named in the stem.
• 2025 Exam 2 Q12b — "Many students answered 115 by including the children who had left the centre." (the absorbing "L" row must be excluded from a "total enrolled" count)
• 2024 GM Exam 2 Q12c — "Some students did not subtract the departed workers and answered 190."
• 2018 Exam 2 Q4a — "Many students did not recognise that the 2700 km of highway remained constant throughout."
• 2017 Exam 2 Q2c — "This question did not require any working if students realised that the original population of 750 remained unchanged throughout the transition process."

3.9 Percentage vs proportion vs decimal vs scalar

• 2021 Exam 2 Q1b — "The correct answer was 1.05. … Students often responded with 0.05." (52 % scored 0)
• 2018 Exam 2 Q2b — "Numbers such as 0.04, −0.01 and −0.02 were often included in students' answers." (should be 1.04, 0.99, 0.98)
• 2006 Exam 2 Q2a — "Some students wrote percentages despite being asked for proportions. … The percentage, 8 %, was sometimes written as a proportion of 0.8."
• 2009 Exam 2 Q3bi — "A common incorrect answer was 0.25 %." (should be 25 %)
• 2017 Exam 2 Q2b — "Some students gave the answers as percentages. A few just gave 40, 60, 20."
• 2007 Exam 2 Q2bii (Core analogue) and 2015 Exam 2 Q3b — "Many students did not convert 21 students into the required percentage."

3.10 Notation and presentation

• Brackets on a 1 × 1 product. 2016 report: "The product of two matrices produces a matrix. The brackets needed to be included." 2017 NHT report: "The brackets must be included."
• Matrix brackets on any hand-drawn matrix. 2007 report: "Matrix brackets were required."
• No commas between elements. 2011 report: "Some students wrote their four elements separated by commas or dots; however, this is not correct mathematical notation."
• Label the matrix as instructed. 2008 report: "The question expected students to label the matrix as shown."
• Arrows on every edge of a directed graph. 2024 GM Exam 2 report: "All edges required arrows otherwise it is no longer a directed graph."
• Show working on ≥2-mark questions. 2025 report: "For questions worth more than one mark, students are advised to show working. An incorrect answer on its own will not be awarded any marks." 2022 report, Q4b: "Many students showed no working for this question." 2017 report, Q1ci: "Even those who did so correctly usually showed no working." 2014 report, Q3b: "This question was poorly answered, with very few instances seen of working out that might earn a method mark."
• Transcription errors from CAS. 2019 report Q1d: "Some responses had transcription errors." 2020 report Q3a: "Most errors for this question appeared to be due to a transcription error from technology." 2024 assessment-guide video: "Some had the first matrix correct, but an incorrect product, which appeared to be from careless calculator use." 2007 report: "Errors in typing the matrix elements into the calculator seemed common. There was little room for method marks here … If an error in data entry had been made, none of these answers would have appeared and thus the student scored zero."
• Don't leave the last page blank. 2007 report, Q2d (50 % scored zero): "Many otherwise strong students left the two questions on this page blank. The 'Turn over' instruction is provided at the bottom of the page to encourage students to read the last page of the examination paper."

3.11 Specific conceptual confusions worth pre-empting

• A "1" on the leading diagonal = absorbing state. 2024 report: "Students who recognised that the 1 in the transition matrix would result in an absorbing state performed well on this question." Only 24 % did.
• Columns of a transition matrix sum to 1 — but a general "attendance" matrix need not. 2010 report Q4b: "It should be noted that the numbers in the columns of G did not add up to one. This meant that, depending upon the initial state matrix, the total weekly number attending the two games could gradually increase or decrease. If G had been a transition matrix with columns adding up to one, then the total numbers attending each week would remain constant regardless of the initial state matrix."
• Pⁿ = I means an n-step cycle. 2025 report Q14c (11 % correct): "Order of the activities rotates on a four-day cycle. There are 10 cycles completed in the 40-day program. Only a small proportion of students showed an understanding of the effect of the identity matrix."
• A "binary matrix" answer is not automatically the identity. 2019 report Q2d (21 % correct): "Many responses were incorrect. Some gave the identity matrix; others gave various assortments of 1s and 0s."
• Two-step communication links are paths, not just a count. 2021 report Q2c (21 % correct): "Alex – Brie – Dex and Alex – Elena – Dex. … Most students appeared not to interpret this question correctly."
• A negative element in B means removal, a positive one means addition. 2017 Exam 1 Q7 report: "At this stage of solution, the question required students to correctly interpret the signs of the elements in matrix B. A positive value means fish are added; a negative value means fish are sold or removed."
• A "symmetric" adjacency matrix must be updated in both the row and the column. 2013 report Q1c: "A common incorrect answer had 0, 1, 1, 1, 0 in the first row, which was unchanged from the original matrix W."
• Mean-of-a-matrix requires a scalar as well as a summing matrix. 2013 Exam 1 Q7, 2023 Exam 2 Q9a, 2024 NHT Exam 2 Q11a.

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4. QUICK REFERENCE — what is essentially guaranteed

Exam 2 (12 marks, 2023+; 12–15 marks, 2006–2022). In every paper in the corpus the matrices block contains at least one of each of:

1. Order / element-notation opener (1 mark, ~90 % success) — §1.1a, §1.1b
2. A product to compute and then interpret in context (1 + 1, interpret half ~40–60 %) — §1.3a
3. "Write a matrix calculation … to show that … is $X" (1 mark, ~60 %) — §1.4b
4. A transition matrix ↔ transition diagram conversion (1–2 marks) — §1.10a
5. Interpret one element of the transition matrix in context (1 mark, ~25–67 %) — §1.10b
6. A backward-percentage question (1 mark, ~17–30 % — the hardest routine mark in the topic) — §1.10d
7. A long-term / steady-state question (1 mark, ~39 %) — §1.10e
8. An Sₙ₊₁ = T Sₙ + B iteration or reverse-iteration (1–2 marks, ~12–39 %) — §1.11b
9. Since 2023: a Leslie matrix or permutation matrix question (3–4 marks) — §1.11, §1.12

Exam 1 (8 MC questions, Q25–32 since 2023). The reliable spread per paper: - 1–2 on matrix arithmetic / scalar / element rule (very high facility) - 1 on whether a product is defined / order of a product - 1 on determinant, inverse or unique solution - 1 on communication matrix or dominance matrix - 1 on permutation matrix (since 2015) - 1–2 on transition matrix, Sₙ = Tⁿ S₀ or long run (consistently the lowest facility: 2020, 2021, 2022 Exam 1 reports all single out the recurrence questions — "Students did not score as highly on questions involving the use of a matrix recurrence relation") - 1 on Leslie matrix (2023+, every paper)

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5. SOURCE FILE INDEX

Papers (matrices sections read in full): Documents_exams_mathematics_2006furmath1-w.txt, ...2006furmath2-w.txt, ...2007furmath1.txt, ...2007furmath2.txt, ...2008furmath1-w.txt, ...2008furmath2-w.txt, ...2009furmath1-w.txt, ...2009furmath2-w.txt, ...2010furmath1-w.txt, ...2010furmath2-w.txt, ...2011furmath1-w.txt (decoded), ...2011furmath2-w.txt (decoded), ...2012_2012furmath1-w.txt, ...2012_2012furmath2-w.txt, ...2013_2013furmath1-w.txt, ...2013_2013furmath2-w.txt, ...2014_2014furmath1-w.txt, ...2014_2014furmath2-w.txt, ...2015_2015furmath1-w.txt, ...2015_2015furmath2-w.txt, ...2016_2016furmath1-w.txt, ...2016_2016furmath2-w.txt, ...2017_2017furmath1-w.txt, ...2017_2017furmath2-w.txt, ...2017_nht_2017FM1-nht-w.txt, ...2017_nht_2017FM2-nht-w.txt, ...2018_2018furmath1-w.txt, ...2018_2018furmath2-w.txt, ...2018_nht_2018FM1-nht-w.txt, ...2018_nht_2018FM2-nht-w.txt, ...2019_2019furmath1-w.txt, ...2019_2019furmath2-w.txt, ...2019_NHT_2019FM1-nht-w.txt, ...2019_NHT_2019FM2-nht-w.txt, ...2020_2020furmath1-w.txt, ...2020_2020furmath2-w.txt, ...2021_2021furmath1-w.txt, ...2021_2021furmath2-w.txt, ...2021_NHT_2021FM1-nht-w.txt, ...2021_NHT_2021FM2-nht-w.txt, ...2022_2022furmath1-w.txt, ...2022_2022furmath2-w.txt, ...2022_NHT_2022furmath1-NHT-w.txt, ...2022_NHT_2022furmath2-nht-w.txt, ...2023_2023genmath1-w.txt, ...2023_2023genmath2-w.txt, ...2023_NHT_2023FM1-nht-w.txt, ...2023_NHT_2023FM2-nht-w.txt, ...2024_NHT_2024GM1-nht-w.txt, ...2024_NHT_2024GM2-nht-w.txt, 2025-11_2025-GeneralMaths1_0.txt, 2025-11_2025-GeneralMaths2.txt, 2026-05_2026-NHT-GeneralMaths1.txt, 2026-06_2026-NHT-GeneralMaths2.txt, Documents_exams_mathematics_genmath1-sample-w.txt, ...genmath2-sample-w.txt, ...generalmaths1-formula-w.txt, ...generalmaths2-formula-w.txt, 2025-04_genmath-specs-w.txt.

Reports / assessment guides: ...furthermaths1_assessrep_06.txt, ...furthermaths2_assessrep_06.txt, ...further1_assessrep_07.txt, ...further2_assessrep_07.txt, ...further_maths1_assessrep_08.txt, ...further_maths2_assessrep_08.txt, ...further1_assessrep_09.txt, ...further2_assessrep_09.txt, ...further1_assessrep_10.txt, ...further2_assessrep_10.txt, ...further1_assessrep_11.txt, ...furthermaths2_assessrep_11.txt, ...2012_FM1_assessrep_12.txt, ...2012_fm2_assessrep12.txt, ...2013_FM1_examrep13.txt, ...2013_FM2_examrep13.txt, ...2014_FM1_examrep14.txt, ...2014_FM2_examrep14.txt, ...2015_FM1_examrep15.txt, ...2015_FM2_examrep15.txt, ...2016_FM1_examrep16.txt, ...2016_FM2_examrep16.txt, ...2017_fm1_examrep17.txt, ...2017_fm2_examrep17.txt, ...2017_nht_fm1nht_examrep17.txt, ...2017_nht_fm2nht_examrep17.txt, ...2018_fm1_examrep18.txt, ...2018_FM2_examrep18.txt, ...2018_nht_furthermaths1nht_examrep18.txt, ...2018_nht_furthermaths2nht_examrep18.txt, ...2019_FM1_examrep19.txt, ...2019_FM2_examrep19.txt, ...2019_NHT_fm1nht_examrep19.txt, ...2019_NHT_fm2nht_examrep19.txt, ...2020_2020furmaths1-exam-report.txt, ...2020_2020furmaths2-exam-report.txt, ...2021_2021furthermath1-report.txt, ...2021_2021furthermath2-report.txt, ...2021_NHT_2021-NHT-Further-Maths-1-exam-report.txt, ...2021_NHT_2021-NHT-Further-Maths-2-exam-report.txt, ...2022_2022furmaths1-report.txt, ...2022_2022furmaths2-report.txt, ...2022_NHT_2022furthmaths1-NHT-report.txt, ...2022_NHT_2022-furthmaths2-NHT-report.txt, ...2023_NHT_2023furthermaths1NHT-report.txt, ...2023_NHT_2023furthermaths2NHT-report.txt, 2025-07_2023generalmaths1-report.txt, 2025-03_2024generalmaths1-report.txt, 2025-03_2024generalmaths2-report.txt, 2025-04_2024generalmathematics2-assessment-guide.txt, 2025-04_General_Maths_Examination_2.txt (assessment-guide video transcript), ...2024_NHT_2024NHTgeneralmaths2-report.txt, 2026-02_2024-NHTgeneralmaths1-report.txt, 2025-06_2025NHT-GeneralMath1-assessment-guide.txt, 2025-06_2025NHT-GeneralMath2-assessment-guide.txt, 2025-10_2025-NHT-general-maths1-report.txt, 2025-10_2025-NHT-general-maths2-report.txt, 2025-11_2025-GeneralMaths1-assessment-guide.txt, 2025-11_2025-GeneralMaths2-assessment-guide.txt, 2026-01_2025-GeneralMaths2-report.txt, 2026-02_2025-GeneralMaths1-report.txt, 2026-06_2026-NHT-GeneralMaths1-assessment-guide_0.txt, 2026-06_2026-NHT-GeneralMaths2-assessment-guide.txt.

Corpus gaps (noted for completeness): Documents_exams_mathematics_2024_2024GeneralMaths1-w.txt, ...2024_2024GeneralMaths2-w.txt, 2025-05_2025-NHT-generalmaths1.txt and 2025-05_2025-NHT-generalmaths2.txt are empty (32 bytes). Claims about those four papers in this document are sourced from their examination reports and assessment guides, not from the papers themselves. The 2011 Exam 1/Exam 2 files use a shifted font encoding and were decoded (+29 ASCII) before quoting.