Matrices and Geometric Transformations Worksheet with Answers

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GCSE · Matrices and Transformations

M2 Matrices and Geometric Transformations

AQA 8365 · Calculator allowed · about 135 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

1

Four matrices are shown below.
A = [[1, 0], [0, -1]]
B = [[-1, 0], [0, 1]]
C = [[0, 1], [1, 0]]
D = [[-1, 0], [0, -1]]
Which matrix represents a reflection in the x-axis?

A) A
B) B
C) C
D) D

(Total for Question 1 is 1 mark)

2

P = [[4, -1], [2, 3]] and Q = [[-2, 5], [1, -4]].

(a)Write down the order of matrix P.(1)

(b)Find P + Q.(2)

(c)Find 2P - Q.(3)

(Total for Question 2 is 6 marks)

3

A = [[3, 0], [-2, 5]] and B = [[1, 4], [6, -3]].

(a)Find 3A - 2B.(3)

(b)X is a matrix such that X + B = A. Find X.(2)

(Total for Question 3 is 5 marks)

4

M = [[2, -1], [3, 4]] and v = [[5], [-2]].

(a)Calculate Mv.(2)

(b)The point (5, -2) is transformed using matrix M. Write down the coordinates of the image point.(1)

(Total for Question 4 is 3 marks)

5

A = [[1, 2], [0, -1]] and B = [[3, 1], [2, 4]].

(a)Calculate AB.(3)

(b)Calculate BA.(3)

(c)State, giving a reason, whether matrix multiplication is commutative.(1)

(Total for Question 5 is 7 marks)

6

N = [[5, 2], [-3, 4]].

(a)Write down the 2 x 2 identity matrix, I.(1)

(b)Calculate NI and IN, and comment on your results.(3)

(Total for Question 6 is 4 marks)

7

A = [[4, 2], [3, 5]] and B = [[6, 3], [4, 2]].

(a)Calculate the determinant of A.(2)

(b)Calculate the determinant of B and hence state, with a reason, whether B is singular.(3)

(Total for Question 7 is 5 marks)

8

C = [[3, 1], [5, 2]].

(a)Find the determinant of C.(1)

(b)Find C^-1.(3)

(Total for Question 8 is 4 marks)

9

A pair of simultaneous equations is given:
3x + y = 11
5x + 2y = 19
These can be written in matrix form Cv = k, where C = [[3, 1], [5, 2]] (the same matrix as in the previous question), v = [[x], [y]] and k = [[11], [19]].

(a)Write down the entries of k.(1)

(b)Given that C^-1 = [[2, -1], [-5, 3]], find the values of x and y.(3)

(Total for Question 9 is 4 marks)

10

Triangle T has vertices A(2, 1), B(4, 1) and C(4, 3).

(a)Write down the matrix that represents a reflection in the x-axis.(1)

(b)Write down the matrix that represents a reflection in the line y = x.(1)

(c)Find the coordinates of the image of triangle T after a reflection in the y-axis.(3)

(Total for Question 10 is 5 marks)

11

Point P has coordinates (3, -2).

(a)Write down the matrix representing a rotation of 90 degrees anticlockwise about the origin.(1)

(b)Write down the matrix representing a rotation of 180 degrees about the origin.(1)

(c)Point P is rotated 90 degrees clockwise about the origin. Find the coordinates of the image.(3)

(Total for Question 11 is 5 marks)

12

A matrix of the form [[k, 0], [0, k]] represents an enlargement, centre the origin, scale factor k.

(a)Write down the matrix that represents an enlargement, centre the origin, scale factor 4.(1)

(b)Under the transformation represented by [[k, 0], [0, k]], the point (2, 3) maps to (-6, -9). Find the value of k.(2)

(c)Describe fully the single transformation represented by the matrix in part (b).(2)

(Total for Question 12 is 5 marks)

13

Three matrices are given:
M1 = [[0, 1], [1, 0]]
M2 = [[-1, 0], [0, -1]]
M3 = [[0, -1], [-1, 0]]

(a)Describe fully the transformation represented by M1.(2)

(b)Describe fully the transformation represented by M2.(2)

(c)Describe fully the transformation represented by M3.(2)

(Total for Question 13 is 6 marks)

14

R = [[0, -1], [1, 0]] represents a rotation of 90 degrees anticlockwise about the origin. S = [[1, 0], [0, -1]] represents a reflection in the x-axis.

(a)Find the single matrix that represents a reflection in the x-axis followed by a rotation of 90 degrees anticlockwise about the origin.(3)

(b)Describe fully the single transformation represented by your answer to part (a).(2)

(Total for Question 14 is 5 marks)

15

Triangle T has area 6 cm² and is transformed by matrix A = [[2, 1], [3, 4]].

(a)Find det(A).(2)

(b)Find the area of the image of T after the transformation.(2)

(c)A second triangle is transformed by matrix B, where det(B) = -3. The image has area 45 cm². Find the area of the original triangle.(2)

(Total for Question 15 is 6 marks)

16

A transformation is represented by matrix T = [[a, b], [c, d]]. Under this transformation, (1, 0) maps to (2, 5) and (0, 1) maps to (-3, 4).

(a)Find the matrix T.(3)

(b)Use your matrix to find the image of the point (-1, 3).(3)

(Total for Question 16 is 6 marks)

17

A transformation is represented by matrix M = [[3, -2], [2, -2]].

(a)Show that the origin is an invariant point under this transformation.(1)

(b)Find M - I, where I is the 2 x 2 identity matrix.(1)

(c)By finding the determinant of M - I, determine whether the transformation has any invariant points other than the origin.(3)

(Total for Question 17 is 5 marks)

18

U = [[0, 1], [1, 0]] represents a reflection in the line y = x. S = [[1, 0], [0, -1]] represents a reflection in the x-axis (as before).

(a)Find the matrix representing a reflection in the line y = x followed by a reflection in the x-axis.(2)

(b)Find the matrix representing a reflection in the x-axis followed by a reflection in the line y = x.(2)

(c)Describe fully the single transformation represented by each of your answers to parts (a) and (b), and state what this shows about combining transformations.(2)

(Total for Question 18 is 6 marks)

19

Transformation A is a rotation of 90 degrees anticlockwise about the origin, represented by matrix R = [[0, -1], [1, 0]]. Transformation B is a reflection in the line y = -x, represented by matrix F = [[0, -1], [-1, 0]].

(a)Find the matrix representing transformation A followed by transformation B.(2)

(b)Describe fully the single transformation equivalent to the combined transformation in part (a).(2)

(c)The point (4, -1) undergoes transformation A followed by transformation B. Use your matrix from part (a) to find the coordinates of the image.(2)

(d)Triangle PQR has area 9 cm². Find the area of its image after the combined transformation in part (a), giving a reason.(2)

(Total for Question 19 is 8 marks)

20

Under a transformation represented by matrix T = [[p, q], [r, s]], the point (2, 1) maps to (7, 4) and the point (1, 3) maps to (6, 2).

(a)Form two pairs of simultaneous equations and find the values of p, q, r and s.(4)

(b)Using your matrix, find the image of the point (4, -2).(2)

(c)Find T^-1 and use it to find the point that maps to the image (13, 6) under transformation T.(5)

(Total for Question 20 is 11 marks)

21

A transformation is represented by matrix M = [[5, -2], [4, -1]]. A line through the origin with equation y = mx is invariant under this transformation.

(a)Show that m satisfies the equation m² - 3m + 2 = 0.(3)

(b)Hence find the equations of the two invariant lines through the origin.(3)

(Total for Question 21 is 6 marks)

Mark scheme · M2 Matrices and Geometric Transformations

Question 1

B1 A cao
Answer: A) A

Question 2

(a) B1 2 x 2 cao
(a) Answer: 2 x 2
(b) M1 adds corresponding entries, with at least two correct
(b) A1 [[2, 4], [3, -1]] cao
(b) Answer: [[2, 4], [3, -1]]
(c) M1 finds 2P = [[8, -2], [4, 6]]
(c) M1 subtracts Q from 2P entrywise
(c) A1 [[10, -7], [3, 10]] cao
(c) Answer: [[10, -7], [3, 10]]

Question 3

(a) M1 finds 3A = [[9, 0], [-6, 15]]
(a) M1 finds 2B = [[2, 8], [12, -6]]
(a) A1 [[7, -8], [-18, 21]] cao
(a) Answer: [[7, -8], [-18, 21]]
(b) M1 rearranges to X = A - B
(b) A1 [[2, -4], [-8, 8]] cao
(b) Answer: [[2, -4], [-8, 8]]

Question 4

(a) M1 correct method (row by column), at least one entry correct
(a) A1 [[12], [7]] cao
(a) Answer: [[12], [7]]
(b) B1 (12, 7) ft from part (a)
(b) Answer: (12, 7)

Question 5

(a) M1 attempts row by column method, at least one correct entry
(a) A1 two correct entries
(a) A1 [[7, 9], [-2, -4]] cao
(a) Answer: [[7, 9], [-2, -4]]
(b) M1 attempts row by column method, at least one correct entry
(b) A1 two correct entries
(b) A1 [[3, 5], [2, 0]] cao
(b) Answer: [[3, 5], [2, 0]]
(c) B1 AB is not equal to BA (compare answers to (a) and (b)), so matrix multiplication is not commutative
(c) Answer: Not commutative, since AB is not equal to BA

Question 6

(a) B1 [[1, 0], [0, 1]] cao
(a) Answer: [[1, 0], [0, 1]]
(b) M1 finds NI or IN correctly
(b) A1 both equal [[5, 2], [-3, 4]]
(b) B1 comment: multiplying any matrix by I, in either order, leaves it unchanged, so I is the multiplicative identity oe
(b) Answer: NI = IN = [[5, 2], [-3, 4]] = N

Question 7

(a) M1 4(5) - 2(3)
(a) A1 14 cao
(a) Answer: 14
(b) M1 6(2) - 3(4)
(b) A1 0 cao
(b) B1 dependent: B is singular because det(B) = 0 oe
(b) Answer: det(B) = 0, so B is singular

Question 8

(a) B1 1 cao
(a) Answer: 1
(b) M1 swaps the leading diagonal entries and negates the other two, forming [[2, -1], [-5, 3]]
(b) M1 divides by det(C)
(b) A1 [[2, -1], [-5, 3]] cao
(b) Answer: [[2, -1], [-5, 3]]

Question 9

(a) B1 [[11], [19]] cao
(a) Answer: [[11], [19]]
(b) M1 sets up v = C^-1 k
(b) M1 correct multiplication
(b) A1 x = 3, y = 2 cao
(b) Answer: x = 3, y = 2

Question 10

(a) B1 [[1, 0], [0, -1]] cao
(a) Answer: [[1, 0], [0, -1]]
(b) B1 [[0, 1], [1, 0]] cao
(b) Answer: [[0, 1], [1, 0]]
(c) M1 uses the reflection matrix [[-1, 0], [0, 1]] or the mapping (x, y) -> (-x, y)
(c) A1 at least two correct image vertices
(c) A1 A'(-2, 1), B'(-4, 1), C'(-4, 3) cao
(c) Answer: A'(-2, 1), B'(-4, 1), C'(-4, 3)

Question 11

(a) B1 [[0, -1], [1, 0]] cao
(a) Answer: [[0, -1], [1, 0]]
(b) B1 [[-1, 0], [0, -1]] cao
(b) Answer: [[-1, 0], [0, -1]]
(c) M1 uses the rotation matrix [[0, 1], [-1, 0]] for a 90 degree clockwise rotation
(c) M1 correct multiplication
(c) A1 (-2, -3) cao
(c) Answer: (-2, -3)

Question 12

(a) B1 [[4, 0], [0, 4]] cao
(a) Answer: [[4, 0], [0, 4]]
(b) M1 sets up 2k = -6 or 3k = -9
(b) A1 k = -3 cao
(b) Answer: k = -3
(c) B1 enlargement
(c) B1 centre the origin, scale factor -3
(c) Answer: Enlargement, centre (0, 0), scale factor -3

Question 13

(a) B1 reflection
(a) B1 in the line y = x
(a) Answer: Reflection in the line y = x
(b) B1 rotation
(b) B1 180 degrees about the origin oe (e.g. enlargement, scale factor -1)
(b) Answer: Rotation of 180 degrees about the origin
(c) B1 reflection
(c) B1 in the line y = -x
(c) Answer: Reflection in the line y = -x

Question 14

(a) M1 forms the product R x S, in this order
(a) M1 correct multiplication attempt
(a) A1 [[0, 1], [1, 0]] cao
(a) Answer: [[0, 1], [1, 0]]
(b) B1 reflection
(b) B1 in the line y = x
(b) Answer: Reflection in the line y = x

Question 15

(a) M1 2(4) - 1(3)
(a) A1 5 cao
(a) Answer: 5
(b) M1 area x |det(A)|
(b) A1 30 cm² cao
(b) Answer: 30 cm²
(c) M1 area_image / |det(B)|
(c) A1 15 cm² cao
(c) Answer: 15 cm²

Question 16

(a) M1 recognises that the first column of T is the image of (1, 0)
(a) M1 recognises that the second column of T is the image of (0, 1)
(a) A1 T = [[2, -3], [5, 4]] cao
(a) Answer: T = [[2, -3], [5, 4]]
(b) M1 sets up T x [[-1], [3]]
(b) M1 correct multiplication
(b) A1 (-11, 7) cao
(b) Answer: (-11, 7)

Question 17

(a) B1 shows M x [[0], [0]] = [[0], [0]] (answer printed)
(a) Answer: M x [[0], [0]] = [[0], [0]] (shown)
(b) B1 [[2, -2], [2, -3]] cao
(b) Answer: [[2, -2], [2, -3]]
(c) M1 2(-3) - (-2)(2)
(c) A1 -2 cao
(c) B1 dependent: since det(M - I) is not 0, the origin is the only invariant point oe
(c) Answer: det(M - I) = -2, so the origin is the only invariant point

Question 18

(a) M1 forms S x U with a correct multiplication attempt
(a) A1 [[0, 1], [-1, 0]] cao
(a) Answer: [[0, 1], [-1, 0]]
(b) M1 forms U x S with a correct multiplication attempt
(b) A1 [[0, -1], [1, 0]] cao
(b) Answer: [[0, -1], [1, 0]]
(c) B1 (a) is a rotation of 90 degrees clockwise about the origin, (b) is a rotation of 90 degrees anticlockwise about the origin
(c) B1 the two matrices are different, so combining transformations is not commutative (the order matters) oe
(c) Answer: (a) rotation of 90 degrees clockwise about the origin; (b) rotation of 90 degrees anticlockwise about the origin; the order of combining transformations matters

Question 19

(a) M1 forms F x R with a correct multiplication attempt
(a) A1 [[-1, 0], [0, 1]] cao
(a) Answer: [[-1, 0], [0, 1]]
(b) B1 reflection
(b) B1 in the y-axis
(b) Answer: Reflection in the y-axis
(c) M1 correct multiplication set up
(c) A1 (-4, -1) cao
(c) Answer: (-4, -1)
(d) M1 |det| = 1, since the combined transformation is a reflection, which preserves area
(d) A1 9 cm², with correct reason
(d) Answer: 9 cm² (area unchanged since |det| = 1)

Question 20

(a) M1 forms 2p + q = 7 and p + 3q = 6, and solves for p and q
(a) A1 p = 3, q = 1
(a) M1 forms 2r + s = 4 and r + 3s = 2, and solves for r and s
(a) A1 r = 2, s = 0
(a) Answer: T = [[3, 1], [2, 0]]
(b) M1 correct multiplication
(b) A1 (10, 8) cao
(b) Answer: (10, 8)
(c) M1 det(T) = 3(0) - 1(2) = -2
(c) M1 forms the inverse using the formula (1/det) x [[s, -q], [-r, p]]
(c) A1 T^-1 = [[0, 0.5], [1, -1.5]] cao
(c) M1 multiplies T^-1 by [[13], [6]]
(c) A1 (3, 4) cao
(c) Answer: T^-1 = [[0, 0.5], [1, -1.5]]; the preimage of (13, 6) is (3, 4)

Question 21

(a) M1 applies M to the general point (x, mx) to get (5x - 2mx, 4x - mx)
(a) M1 sets the image's y-coordinate equal to m times its x-coordinate, 4x - mx = m(5x - 2mx), and divides by x
(a) A1 correctly rearranges to m² - 3m + 2 = 0 (answer printed)
(a) Answer: m² - 3m + 2 = 0 (shown)
(b) M1 factorises (m - 1)(m - 2) = 0 or uses the quadratic formula
(b) A1 m = 1, giving y = x
(b) A1 m = 2, giving y = 2x
(b) Answer: y = x and y = 2x

Matrices and Geometric Transformations - Worksheets, Questions and Revision

M2 Matrices and Geometric Transformations

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Question 2

Question 3

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Question 7

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Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18

Question 19

Question 20

Question 21

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Question 11

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Question 21