Matrices and Geometric Transformations - Worksheets, Questions and Revision

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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

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

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