Core Pure: Matrices and Transformations Depth - Worksheets, Questions and Revision

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A-Level · Further Pure Mathematics - Core Pure

FP.CP13 Core Pure: Matrices and Transformations Depth

EDEXCEL 9FM0 · Calculator allowed · about 130 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

The matrices A and B are defined by A = [[2,-1],[3,4]] and B = [[1,0],[-2,5]].

(a)Find the matrix AB.(2)

(b)Find the matrix BA.(2)

(c)State, with a reason, whether AB = BA.(1)

(Total for Question 1 is 5 marks)

The matrix M is defined by M = [[4,7],[1,2]].

(a)Find det(M).(2)

(b)Find M^-1.(2)

(c)Hence use a matrix method to solve the simultaneous equations 4x + 7y = 15 and x + 2y = 4.(3)

(Total for Question 2 is 7 marks)

The matrix C is defined by C = [[1,2,3],[0,4,5],[1,0,6]]. Find det(C), showing your method clearly.

(Total for Question 3 is 4 marks)

The matrix R represents a rotation about the origin O through 60 degrees anticlockwise, so that R = [[cos60,-sin60],[sin60,cos60]].

(a)Write R in exact (surd) form.(2)

(b)Find the exact image of the point (4,0) under R.(2)

(c)State the value of det(R) and interpret this value geometrically for the transformation R.(2)

(Total for Question 4 is 6 marks)

The transformation T is represented by the matrix N = [[5,4],[1,2]].

(a)Find the equations of the two invariant lines of T that pass through the origin.(6)

(b)Determine which of the two invariant lines found in part (a) consists entirely of invariant points (that is, every point on the line maps to itself).(2)

(Total for Question 5 is 8 marks)

The matrix A = [[2,0],[0,2]] represents an enlargement, centre O, scale factor 2. The matrix B = [[0,-1],[1,0]] represents a rotation of 90 degrees anticlockwise about O. The enlargement is applied first, followed by the rotation.

(a)Find the single matrix, BA, that represents the combined transformation.(3)

(b)Use your matrix from part (a) to find the image of the point (3,-1) under the combined transformation.(2)

(c)Describe the single geometrical transformation equivalent to the matrix BA.(1)

(Total for Question 6 is 6 marks)

The matrix P is defined by P = [[1,2,1],[0,1,-1],[2,3,0]].

(a)Show that det(P) = -3.(3)

(b)Find P^-1.(5)

(c)Hence solve the system of equations: x + 2y + z = 3, y - z = -6, 2x + 3y = 0.(2)

(Total for Question 7 is 10 marks)

The matrix T is defined by T = [[3,-1],[2,4]]. Triangle OAB has vertices O(0,0), A(2,0) and B(0,3).

(a)Find det(T).(2)

(b)State the area of triangle OAB.(1)

(c)Find the area of the image of triangle OAB under the transformation represented by T.(2)

(d)The transformation represented by T is applied twice in succession. Find the area scale factor of this combined transformation.(2)

(Total for Question 8 is 7 marks)

The matrix Q is defined by Q = [[2k,1],[3,k+4]], where k is a constant.

(a)Given that y = x is an invariant line of the transformation represented by Q, show that k = 6.(4)

(b)Using this value of k, find the equation of the other invariant line of the transformation that passes through the origin.(6)

(Total for Question 9 is 10 marks)

The matrix A = [[2,a],[b,3]], where a and b are positive constants.

(a)Show that A² = [[4+ab,5a],[5b,ab+9]].(3)

(b)Given that A² = [[7,15],[5,12]], find the values of a and b.(5)

(Total for Question 10 is 8 marks)

The matrix M(k) is defined by M(k) = [[k,1],[2,k]], where k is a real constant.

(a)Find det(M(k)) in terms of k.(2)

(b)Find the value(s) of k for which M(k) is singular.(2)

(c)When k = 3, find M(3)^-1.(3)

(d)Show that the only invariant point of the transformation represented by M(3) is the origin.(3)

(e)The transformation represented by M(3) is applied to the point (1,-2), followed by a translation by the vector (3,1). Find the coordinates of the final image point.(2)

(Total for Question 11 is 12 marks)

In three dimensions, a rotation of angle θ anticlockwise about the z-axis (viewed from the positive z-axis looking towards the origin) is represented by the matrix Rz(θ) = [[cos(θ),-sin(θ),0],[sin(θ),cos(θ),0],[0,0,1]].

(a)Write down Rz(90), the matrix representing a rotation of 90 degrees anticlockwise about the z-axis.(2)

(b)Find the image of the point (2,5,-3) under this rotation.(2)

(c)State the equation(s) of the invariant line of this transformation (the line consisting entirely of invariant points).(2)

(d)Find det(Rz(90)) and interpret its value geometrically.(3)

(Total for Question 12 is 9 marks)

A system of three equations in x, y and z is given by: x + 2y - z = 4; 2x - y + 3z = 3; 4x + 3y + z = k, where k is a constant.

(a)Show that the determinant of the coefficient matrix [[1,2,-1],[2,-1,3],[4,3,1]] is 0.(3)

(b)By finding a linear combination of the first two equations that produces the same left-hand side as the third equation, find the value of k for which the system is consistent.(4)

(c)For this value of k, find a vector equation of the line in which the three planes intersect.(4)

(Total for Question 13 is 11 marks)

Mark scheme · FP.CP13 Core Pure: Matrices and Transformations Depth

Question 1

(a) M1 attempts at least one correct row-by-column entry of AB
(a) A1 AB = [[4,-5],[-5,20]] cao
(a) Answer: AB = [[4,-5],[-5,20]]
(b) M1 attempts at least one correct row-by-column entry of BA
(b) A1 BA = [[2,-1],[11,22]] cao
(b) Answer: BA = [[2,-1],[11,22]]
(c) B1 states AB is not equal to BA, referencing that matrix multiplication is not commutative in general (e.g. comparing the (1,1) entries 4 and 2)
(c) Answer: AB does not equal BA; matrix multiplication is not commutative in general.

Question 2

(a) M1 attempts ad - bc, i.e. 4(2) - 7(1)
(a) A1 det(M) = 1 cao
(a) Answer: det(M) = 1
(b) M1 uses M^-1 = (1/det(M)) [[d,-b],[-c,a]] with correct entries from M
(b) A1 M^-1 = [[2,-7],[-1,4]] cao
(b) Answer: M^-1 = [[2,-7],[-1,4]]
(c) M1 writes the system as M(x,y)^T = (15,4)^T and multiplies both sides by M^-1
(c) A1 x = 2
(c) A1 y = 1
(c) Answer: x = 2, y = 1

Question 3

M1 expands the determinant along any row or column with correct cofactor structure
A1 correct value for at least two of the three 2x2 minors used, e.g. 4x6-5x0=24, 0x6-5x1=-5, 0x0-4x1=-4
M1 combines the minors with the correct signs and coefficients
A1 det(C) = 22 cao
Answer: det(C) = 22

Question 4

(a) M1 uses cos60 = 1/2 and sin60 = √3/2
(a) A1 R = [[1/2,-√3/2],[√3/2,1/2]] oe cao
(a) Answer: R = [[1/2,-√3/2],[√3/2,1/2]]
(b) M1 multiplies R by the column vector (4,0)^T
(b) A1 image = (2, 2sqrt(3)) oe cao
(b) Answer: (2, 2sqrt(3))
(c) B1 det(R) = 1
(c) B1 correct interpretation: the transformation preserves area (area scale factor 1) and preserves orientation, consistent with R being a rotation
(c) Answer: det(R) = 1; areas are preserved (scale factor 1) and orientation is preserved, as expected for a rotation.

Question 5

(a) M1 considers a general point (1,m) on a line y = mx through O and finds its image under N, i.e. (5+4m, 1+2m)
(a) M1 sets up the invariance condition (1+2m)/(5+4m) = m, or equivalent
(a) A1 correct quadratic 4m² + 3m - 1 = 0 oe
(a) M1 solves the quadratic by a valid method (factorising or the formula)
(a) A1 m = 1/4
(a) A1 m = -1, giving invariant lines y = x/4 and y = -x
(a) Answer: y = x/4 and y = -x
(b) M1 tests a general point (x,-x) on y=-x under N: N(x,-x)^T = (5x-4x, x-2x) = (x,-x)
(b) A1 concludes y = -x consists entirely of invariant points (every point maps to itself), unlike y = x/4, where points map to other points on the same line
(b) Answer: y = -x consists entirely of invariant points.

Question 6

(a) M1 attempts the matrix product BA (rotation matrix multiplied by enlargement matrix, in the correct order)
(a) A1 at least one row of BA correct
(a) A1 BA = [[0,-2],[2,0]] cao
(a) Answer: BA = [[0,-2],[2,0]]
(b) M1 multiplies BA by the column vector (3,-1)^T
(b) A1 image = (2,6) cao
(b) Answer: (2,6)
(c) B1 describes BA as an enlargement, centre O, scale factor 2, combined with a rotation of 90 degrees anticlockwise about O; accept equivalent correct descriptions of this spiral similarity
(c) Answer: A rotation of 90 degrees anticlockwise about O combined with an enlargement, centre O, scale factor 2.

Question 7

(a) M1 attempts expansion of det(P) along any row or column with correct cofactor structure
(a) A1 correct 2x2 minors used, e.g. (1x0-(-1)x3)=3, (0x0-(-1)x2)=2, (0x3-1x2)=-2
(a) A1 det(P) = -3 cso
(a) Answer: det(P) = -3
(b) M1 attempts at least four of the nine cofactors of P
(b) A1 correct cofactor matrix [[3,-2,-2],[3,-2,1],[-3,1,1]]
(b) M1 transposes the cofactor matrix to form the adjugate
(b) A1 adjugate = [[3,3,-3],[-2,-2,1],[-2,1,1]]
(b) A1 P^-1 = (1/-3)[[3,3,-3],[-2,-2,1],[-2,1,1]] = [[-1,-1,1],[2/3,2/3,-1/3],[2/3,-1/3,-1/3]] cao (unsimplified equivalent forms accepted)
(b) Answer: P^-1 = [[-1,-1,1],[2/3,2/3,-1/3],[2/3,-1/3,-1/3]]
(c) M1 multiplies P^-1 by the column vector (3,-6,0)^T
(c) A1 x = 3, y = -2, z = 4 cao
(c) Answer: x = 3, y = -2, z = 4

Question 8

(a) M1 attempts 3(4) - (-1)(2)
(a) A1 det(T) = 14 cao
(a) Answer: det(T) = 14
(b) B1 area = 3
(b) Answer: 3
(c) M1 multiplies the area of OAB by |det(T)|
(c) A1 area = 42 cao
(c) Answer: 42
(d) M1 uses the area scale factor of T² equal to (det T)²
(d) A1 196 cao
(d) Answer: 196

Question 9

(a) M1 finds the image of the point (1,1) under Q, i.e. (2k+1, k+7)
(a) M1 uses the condition that the two coordinates of the image must be equal, since the point lies on y = x
(a) A1 correct equation 2k+1 = k+7 oe
(a) A1 k = 6 cso
(a) Answer: k = 6
(b) M1 substitutes k=6 to get Q = [[12,1],[3,10]] and finds the image of a general point (1,m), i.e. (12+m, 3+10m)
(b) A1 correct image expression
(b) M1 sets up the invariance condition (3+10m)/(12+m) = m
(b) A1 correct quadratic m² + 2m - 3 = 0 oe
(b) M1 solves the quadratic by a valid method, e.g. factorises as (m+3)(m-1) = 0
(b) A1 m = -3 (rejecting the already-known solution m=1), so the other invariant line is y = -3x
(b) Answer: y = -3x

Question 10

(a) M1 attempts the matrix product A x A
(a) M1 at least two of the four entries correct
(a) A1 all four entries correct, matching [[4+ab,5a],[5b,ab+9]] cso
(a) Answer: A² = [[4+ab,5a],[5b,ab+9]]
(b) M1 equates 5a = 15 (from the (1,2) entries)
(b) A1 a = 3
(b) M1 equates 5b = 5 (from the (2,1) entries)
(b) A1 b = 1
(b) B1 verifies using the remaining entries, e.g. 4+ab = 4+3 = 7 and ab+9 = 3+9 = 12, both consistent with the given A²
(b) Answer: a = 3, b = 1

Question 11

(a) M1 attempts k(k) - 1(2)
(a) A1 det(M(k)) = k² - 2 cao
(a) Answer: k² - 2
(b) M1 sets k² - 2 = 0
(b) A1 k = √2 or k = -√2 (both required)
(b) Answer: k = √2 or k = -√2
(c) M1 finds det(M(3)) = 9-2 = 7
(c) M1 uses M^-1 = (1/det)[[d,-b],[-c,a]] with correct entries
(c) A1 M(3)^-1 = (1/7)[[3,-1],[-2,3]] cao
(c) Answer: M(3)^-1 = (1/7)[[3,-1],[-2,3]]
(d) M1 sets up (M(3) - I)(x,y)^T = (0,0)^T, i.e. the equations 2x+y=0 and 2x+2y=0
(d) M1 solves the equations simultaneously (e.g. subtracting to get y=0, hence x=0), or argues via det(M(3)-I) ≠ 0
(d) A1 concludes x=0, y=0 is the only solution, so the only invariant point is the origin cso
(d) Answer: The origin (0,0) is the only invariant point.
(e) M1 finds M(3)(1,-2)^T = (1,-4) and adds the translation vector (3,1)
(e) A1 (4,-3) cao
(e) Answer: (4,-3)

Question 12

(a) M1 uses cos(90) = 0 and sin(90) = 1
(a) A1 Rz(90) = [[0,-1,0],[1,0,0],[0,0,1]] cao
(a) Answer: Rz(90) = [[0,-1,0],[1,0,0],[0,0,1]]
(b) M1 multiplies Rz(90) by the column vector (2,5,-3)^T
(b) A1 (-5,2,-3) cao
(b) Answer: (-5,2,-3)
(c) B1 identifies the invariant line as the z-axis
(c) B1 gives its equation as x=0, y=0 (z takes any value) oe
(c) Answer: The z-axis: x = 0, y = 0 (z takes any value).
(d) M1 attempts the 3x3 determinant expansion of Rz(90)
(d) A1 det(Rz(90)) = 1
(d) B1 interprets: the transformation preserves volume (volume scale factor 1) and preserves orientation, consistent with Rz(90) being a rotation
(d) Answer: det(Rz(90)) = 1; volumes are preserved and orientation is preserved, as expected for a rotation.