plumiMatrices: AQA A‑Level Further Maths questions
Every AQA A-Level Further Maths exam question on C Matrices, 2019–2025, with its mark scheme and examiner’s report.
Paper
All papers
Exam
All years
All 56 questions on Matrices, as a list
- June 2025, Paper 1, Q12 (13 marks): (a) Find the eigenvalues and corresponding eigenvectors of the matrix = 19 3 1 3 11 20 M [5 marks] 12 (b) State, with a reason, the Cartesian equati
- June 2025, Paper 1, Q15 (9 marks): Three planes have equations 2 9 3 3 3 4 x y z x y z t x y z t + − = − + = + + = where t is a constant. The planes meet along a line of intersection. 15 (a) Find
- June 2025, Paper 2, Q2 (1 mark): The quadrilateral Q1 has an area of 5 cm2 The matrix − 4 1 2 1 represents the transformation T The transformation T acts on Q1 to give the quadrilat
- June 2025, Paper 2, Q7 (3 marks): The matrix A is defined by − = − 4 2 6 3 A Find a non-zero 2 × 2 matrix B such that AB = 0 [3 marks]
- June 2025, Paper 2, Q13 (10 marks): The matrix M is defined by − = 1 3 3 1 M 13 (a) The matrix M represents an anticlockwise rotation about the origin through an angle θ, where 0 ≤
- June 2025, AS Paper 1, Q10 (8 marks): The 3× 3 matrix M represents a reflection in the plane z = 0 The point A has position vector 2 5 4 The point A′ is the image of A under the reflection represent
- June 2025, AS Paper 1, Q11 (9 marks): The 2 × 2 matrix A represents an anticlockwise rotation of 150° about the origin. 11 (a) Find matrix A Write each element in its simplest exact form. [2 marks]
- June 2025, AS Paper 1, Q14 (8 marks): (a) The matrices A and B are given by A = [ 4 –3] –1 1 and B = [ 5 4 ] –3 –2 14 (a) (i) Find the matrices A–1 and B–1 [2 marks] 14 (a) (ii) Hence verify that (A
- June 2024, Paper 1, Q12 (10 marks): The line L1 has equation r = 4 2 1 + λ 1 3 –1 The transformation T is represented by the matrix 2 1 0 3 4 6 –5 2 –3 The transformation T transforms the line L1
- June 2024, Paper 2, Q1 (1 mark): It is given that 2 1 3 5 λ –6 = 0 where λ is a constant. Find the value of λ Circle your answer. [1 mark] –28 –8 8 28
- June 2024, Paper 2, Q7 (4 marks): The matrices A and B are defined as follows. A = p – 2 p – 1 0 1 B = 1 2p – 1 0 4 – p Find the values of p such that A and B are commutative under matrix multip
- June 2024, Paper 2, Q10 (4 marks): The matrix C is defined by C = 3 2 –4 5 Prove that the transformation represented by C has no invariant lines of the form y = kx [4 marks]
- June 2024, Paper 2, Q12 (5 marks): The transformation S is represented by the matrix M = 1 –6 2 7 The transformation T is a reflection in the line y = x√ 3 and is represented by the matrix N The
- June 2024, Paper 2, Q14 (10 marks): The matrix M is defined as M = 5 2 1 6 3 2k + 3 2 1 5 where k is a constant. 14 (a) Given that M is a non-singular matrix, find M–1 in terms of k [5 marks] 14 (
- June 2024, AS Paper 1, Q3 (1 mark): The matrix A is such that det(A) = 2 Determine the value of det(A–1) Circle your answer. [1 mark] –2 –1 2 1 2 2
- June 2024, AS Paper 1, Q11 (3 marks): The matrices A and B are given by A = 3i –2 a –i and B = 4 5 –2i –1 where a is a real number. Calculate the product AB in terms of a Give your answer in its sim
- June 2024, AS Paper 1, Q14 (10 marks): The matrix M represents the transformation T, and is given by M = 3 –1 –2 6 14 (a) The point A has coordinates (4, –5) Find the coordinates of the image of A un
- June 2023, Paper 1, Q3 (1 mark): The matrix A = 1 2 0 1 represents a transformation. Which one of the points below is an invariant point under this transformation? Circle your answer. [1 mark]
- June 2023, Paper 1, Q6 (11 marks): The matrix M is given by M = 1 10 a a −6 0 10 0 9 14 −13 2 664 3 775 where a is a real number. The vectors v1, v2, and v3 are eigenvectors of M The correspondin
- June 2023, Paper 1, Q10 (12 marks): The matrix M is defined as M = 2 −1 1 −1 −1 −2 1 2 c 2 64 3 75 where c is a real number. 10 (a) The linear transformation T is represented by the matrix M Show
- June 2023, Paper 2, Q3 (1 mark): The determinant A = 1 1 1 2 0 2 3 2 1 Which one of the determinants below has a value which is not equal to the value of A? Tick (3) one box. [1 mark] 3 1 3 2 0
- June 2023, Paper 2, Q5 (5 marks): Josh and Zoe are solving the following mathematics problem: The curve C1 has equation x2 16 − y2 9 = 1 The matrix M = 0 1 1 0 " # maps C1 onto C2 Find the equat
- June 2023, Paper 2, Q8 (6 marks): A is a non-singular 2 ≤ 2 matrix and AT is the transpose of A 8 (a) Using the result (AB)T = BTAT show that (A−1)T = (AT)−1 [3 marks] 8 (b) It is given that A =
- June 2023, AS Paper 1, Q3 (1 mark): The matrices A and B are given by A = [ 3 1] 0 5 B = [ 0 4] 7 1 Calculate AB Circle your answer. [1 mark] [ 3 5] 7 6 [ 0 20] 21 12 [ 0 4] 0 5 [ 7 13] 35 5
- June 2023, AS Paper 1, Q9 (11 marks): The matrix M represents the transformation T and is given by M = [ 3p + 1 p + 2 12 ] p2 – 3 9 (a) In the case when p = 0 show that the image of the point (4, 5)
- June 2022, Paper 1, Q4 (1 mark): The vector v is an eigenvector of the matrix N with corresponding eigenvalue 4 The vector v is also an eigenvector of the matrix M with corresponding eigenvalue
- June 2022, Paper 1, Q7 (9 marks): The matrix M is defined as M = 1 7 −3 3 6 k + 1 1 3 2 2 664 3 775 where k is a constant. 7 (a) (i) Given that M is a non-singular matrix, find M−1 in terms of k
- June 2022, Paper 1, Q9 (14 marks): Roberto is solving this mathematics problem: The curve C1 has polar equation r2 = 9 sin 2y for all possible values of y Find the area enclosed by C1 Roberto’s s
- June 2022, Paper 2, Q11 (9 marks): (a) Find the eigenvalues and corresponding eigenvectors of the matrix M = 5 2 −3 2 −3 2 13 2 2 64 3 75 [5 marks] 11 (b) (i) Describe how the directions of the i
- June 2022, Paper 2, Q13 (16 marks): (a) The matrix A represents a reflection in the line y = mx , where m is a constant. Show that A = 1 m2 + 1 1 − m2 2m 2m m2 − 1 " # You may use the result in th
- June 2022, AS Paper 1, Q3 (1 mark): Which of the following transformations is represented by the matrix 1 0 0 0 −1 0 0 0 1 2 64 3 75 ? Tick (3) one box. [1 mark] Rotation of 180° about the x-axis
- June 2022, AS Paper 1, Q6 (5 marks): The matrix A is given by A = 5 2 −3 4 6 (a) Find det A [1 mark] 6 (b) Find A−1 [1 mark] 6 (c) Given that AB = 9 6 5 12 and M = 2A + B find the matrix M [3 marks
- June 2022, AS Paper 1, Q11 (4 marks): Prove by induction that, for all integers n × 1 , (ABA−1)n = ABnA−1 where A and B are square matrices of equal dimensions, and A is non-singular. [4 marks]
- November 2021, Paper 1, Q5 (5 marks): The matrix M is defined by M = 3 2 −2 0 1 0 0 0 1 2 64 3 75 Prove by induction that Mn = 3n 3n − 1 −3n + 1 0 1 0 0 0 1 2 64 3 75 for all integers n × 1 [5 marks
- November 2021, Paper 1, Q12 (14 marks): The matrix A = 1 5 3 4 −2 p 8 5 −11 2 64 3 75 , where p is a constant. 12 (a) Given that A is a non-singular matrix, find A−1 in terms of p. State any restricti
- November 2021, Paper 1, Q13 (3 marks): The transformation S is represented by the matrix 3 0 0 1 The transformation T is a translation by the vector 0 −5 Kamla transforms the graphs of various functi
- November 2021, Paper 2, Q1 (1 mark): Which of the following matrices is singular? Circle your answer. [1 mark] 1 0 0 1 1 1 2 2 0 1 1 0 1 −2 1 2
- November 2021, Paper 2, Q11 (9 marks): The Cartesian equation of the line L1 is x + 1 3 = −y + 5 2 = 2z + 5 3 The Cartesian equation of the line L2 is 2x − 1 2 = y − 14 m = z + 12 p The non-singular
- November 2021, AS Paper 1, Q3 (1 mark): The matrix M represents a rotation about the x-axis. M = 1 0 0 0 a 3 p 2 0 b − 1 2 2 666664 3 777775 Which of the following pairs of values is correct? Tick (3)
- November 2021, AS Paper 1, Q4 (1 mark): The point (2, −1) is invariant under the transformation represented by the matrix N Which of the following matrices could be N? Circle your answer. [1 mark] 4 6
- November 2021, AS Paper 1, Q10 (8 marks): Matrix A is given by A = 3 i − 1 i 2 10 (a) Show that det A = a + i where a is an integer to be determined. [2 marks] 10 (b) Matrix B is given by B = 14 − 2i b
- November 2020, Paper 1, Q2 (1 mark): Which one of the matrices below represents a rotation of 90° about the x-axis? Circle your answer. [1 mark] 1 0 0 0 1 0 0 0 −1 2 64 3 75 −1 0 0 0 1 0 0 0 1 2 64
- November 2020, Paper 1, Q7 (7 marks): Three planes have equations (4k + 1)x − 3y + (k − 5)z = 3 (k − 1)x + (3 − k)y + 2z = 1 7x − 3y + 4z = 2 7 (a) The planes do not meet at a unique point. Show tha
- November 2020, Paper 2, Q4 (3 marks): The matrices A and B are defined as follows: A = x + 1 2 x + 2 ×3 " # B = x × 4 x × 2 0 ×2 " # Show that there is a value of x for which AB = kI , where I is th
- November 2020, Paper 2, Q8 (9 marks): (a) Factorise 2a + b + x x + b x2 + b2 0 a ×a2 a + b b b2 as fully as possible. [6 marks] 8 (b) The matrix M is defined by M = 13 + x x + 3 x2 + 9 0 5 ×25 8 3 9
- November 2020, Paper 2, Q9 (7 marks): The matrix C = a ×b b a , where a and b are positive real numbers, and C2 = p3 2 × 1 2 1 2 p3 2 2 6664 3 7775 Use C to show that cos p 12 can be written in the
- November 2020, AS Paper 1, Q4 (5 marks): The matrices A and B are such that A = 2 a 3 0 −2 1 and B = 1 −3 −2 4a 0 5 2 64 3 75 4 (a) Find the product AB in terms of a. [2 marks] 4 (b) Find the determina
- November 2020, AS Paper 1, Q6 (2 marks): Anna has been asked to describe the transformation given by the matrix 1 0 0 0 − 3 p 2 − 1 2 0 1 2 − 3 p 2 2 66664 3 77775 She writes her answer as follows: The
- November 2020, AS Paper 1, Q16 (4 marks): A and B are non-singular square matrices. 16 (a) Write down the product AA−1 as a single matrix. [1 mark] 16 (b) M is a matrix such that M = AB . Prove that M−1
- June 2019, Paper 1, Q7 (4 marks): Three non-singular square matrices, A, B and R are such that AR = B The matrix R represents a rotation about the z-axis through an angle y and B = − cos y sin y
- June 2019, Paper 1, Q9 (9 marks): (a) Solve the equation z3 = 2 p − 6 p i , giving your answers in the form reiy where r > 0 and 0 × y < 2p [5 marks] 9 (b) The transformation represented by the
- June 2019, Paper 1, Q12 (8 marks): Three planes have equations 4x − 5y + z = 8 3x + 2y − kz = 6 (k − 2)x + ky − 8z = 6 where k is a real constant. The planes do not meet at a unique point. 12 (a)
- June 2019, Paper 2, Q9 (13 marks): (a) Find the eigenvalues and corresponding eigenvectors of the matrix M = 1 5 2 5 −3 5 13 10 2 6664 3 7775 [5 marks] 9 (b) Find matrices U and D such that D is
- June 2019, AS Paper 1, Q1 (1 mark): Which of the following matrices is an identity matrix? Circle your answer. [1 mark] 1 1 1 1 1 0 0 1 0 1 1 0 0 0 0 0
- June 2019, AS Paper 1, Q2 (1 mark): Which of the following expressions is the determinant of the matrix a 2 b 5 ? Circle your answer. [1 mark] 5a − 2b 2a − 5b 5b − 2a 2b − 5a
- June 2019, AS Paper 1, Q12 (12 marks): The matrix A is given by A = 1 2 0 3 12 (a) Prove by induction that, for all integers n ≤ 1, An = 1 3n − 1 0 3n [4 marks] 12 (b) Find all invariant lines under
Questions, mark schemes and examiner’s reports are © AQA, from their published past papers. Plumi isn’t affiliated with any exam board. Want whole papers? Every AQA A-Level Further Maths paper.
Rather write your answers by hand on iPad? Plumi marks them against the same mark schemes.