Matrices (A Level Further Maths) - Worksheets, Questions and Revision

12 original exam-style questions - 4 pages of questions with a full mark scheme - free printable PDF.

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A-Level · Further Pure Mathematics - Core Pure

FP.CP2 Matrices

EDEXCEL 9FM0 · Calculator allowed · about 135 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Matrices A and B are given by A = [[2,-1],[3,4]] and B = [[0,5],[-2,1]].

(a)Find A + B.(1)

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

(c)Find the matrix product AB.(3)

(d)Find the matrix product BA and hence state, with a reason, whether matrix multiplication is commutative for A and B.(3)

(Total for Question 1 is 9 marks)

M = [[3,2],[5,4]].

(a)Find det(M).(1)

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

(c)Hence use matrices to solve the simultaneous equations 3x + 2y = 8 and 5x + 4y = 14.(3)

(Total for Question 2 is 6 marks)

R is the matrix representing a reflection in the line y = x. S is the matrix representing a rotation of 90 degrees anticlockwise about the origin O.

(a)Write down the matrix R.(1)

(b)Write down the matrix S.(1)

(c)Find the matrix RS, representing rotation by S followed by reflection by R.(3)

(d)Describe fully the single geometrical transformation represented by RS, and find the image of the point (3,-2) under this transformation.(2)

(Total for Question 3 is 7 marks)

A transformation of the plane is represented by the matrix M = [[5,-2],[4,-1]].

(a)Show that every point on the line y = 2x is an invariant point of the transformation represented by M.(3)

(b)Find the equation of the other invariant line of M through the origin (not consisting entirely of invariant points).(4)

(c)State the scale factor of enlargement associated with this second invariant line.(1)

(d)Verify your answer to part (b) by finding the image of the point (1,1) under M and confirming it lies on the same invariant line, stating the scale factor this confirms.(2)

(Total for Question 4 is 10 marks)

P = [[k,1,1],[1,k,1],[1,1,k]], where k is a real constant.

(a)Show that det(P) = k³ - 3k + 2.(4)

(b)Show that det(P) = (k-1)²(k+2).(2)

(c)Find the value(s) of k for which P is singular.(2)

(Total for Question 5 is 8 marks)

A fairground in Weymouth charges different prices, in pounds, for adult (a), child (c) and senior (s) tickets. Ticket sales over three days are recorded as: Day 1: 2 adult + 3 child + 1 senior = 34 pounds. Day 2: 1 adult + 2 child + 2 senior = 24 pounds. Day 3: 3 adult + 1 child + 1 senior = 32 pounds.

(a)Write this information as a matrix equation Qv = r, where v = (a,c,s)^T, stating Q and r.(1)

(b)Find det(Q).(3)

(c)Find Q^-1.(4)

(d)Hence find the price of an adult, a child and a senior ticket.(3)

(Total for Question 6 is 11 marks)

A transformation of the plane is represented by the matrix T = [[3,1],[1,2]]. Triangle ABC has vertices A(0,0), B(4,0), C(0,2) and is mapped to triangle A'B'C' by T.

(a)Find det(T).(1)

(b)State the area scale factor of the transformation, and hence find the area of triangle A'B'C'.(2)

(c)Find the coordinates of A', B' and C'.(3)

(d)Verify your answer to part (b) by calculating the area of A'B'C' directly, using the coordinates found in part (c).(3)

(Total for Question 7 is 9 marks)

V = [[1,-1],[1,1]].

(a)Show that V can be written as k*R(θ), where R(θ) = [[cos(θ),-sin(θ)],[sin(θ),cos(θ)]], stating the value of k and the value of θ (0 < θ < 90 degrees).(4)

(b)Describe fully the single geometrical transformation represented by V.(2)

(c)Find V² and describe the single geometrical transformation it represents.(4)

(d)Find the image of the point (3,1) under the transformation represented by V².(2)

(Total for Question 8 is 12 marks)

A system of equations is given by x + 2y - z = 3, 2x + y + z = 5, 3x + 3y + (λ)z = μ, where λ and μ are real constants. The coefficient matrix is A(λ) = [[1,2,-1],[2,1,1],[3,3,λ]].

(a)Show that det(A(λ)) = -3*λ.(3)

(b)State the value of λ for which the system does not have a unique solution.(1)

(c)When λ = 0, find the value of μ for which the system is consistent.(3)

(d)Describe geometrically the configuration of the three planes when λ = 0, in the two cases (i) μ = 8 and (ii) μ is not equal to 8.(3)

(Total for Question 9 is 10 marks)

A = [[4,1],[2,3]].

(a)Find A² by direct matrix multiplication.(3)

(b)Given that A satisfies its own characteristic equation, show that A² = 7A - 10I.(2)

(c)By multiplying the result in part (b) by A^-1, show that A^-1 = 0.7I - 0.1A.(3)

(d)Hence write A^-1 as a matrix with fractional entries.(1)

(Total for Question 10 is 9 marks)

A transformation of the plane is represented by the matrix W = [[7,-6],[3,-2]].

(a)Find the eigenvalues of W by solving det(W - λ*I) = 0.(4)

(b)By considering a general line y = mx through the origin, find the equations of the two invariant lines of the transformation represented by W.(5)

(c)Determine which of the two invariant lines consists entirely of invariant points, justifying your answer using the eigenvalues from part (a).(2)

(Total for Question 11 is 11 marks)

A 3D design scaling transformation for a manufactured component is represented by the matrix X = [[2,0,1],[0,1,0],[1,0,2]], applied to position vectors (x,y,z)^T measured in cm.

(a)Find det(X) and interpret its value in the context of the transformation.(3)

(b)A component originally has volume 5 cm³ before the transformation is applied. Find the volume of the transformed component.(1)

(c)Find X^-1.(5)

(d)A transformed point has coordinates (5,2,4). Find its original coordinates before the transformation was applied.(3)

(Total for Question 12 is 12 marks)

Mark scheme · FP.CP2 Matrices

Question 1

(a) B1 A + B = [[2,4],[1,5]] cao
(a) Answer: A + B = [[2,4],[1,5]]
(b) M1 3A = [[6,-3],[9,12]] or 2B = [[0,10],[-4,2]] found correctly
(b) A1 3A - 2B = [[6,-13],[13,10]] cao
(b) Answer: 3A - 2B = [[6,-13],[13,10]]
(c) M1 attempt at matrix multiplication with at least one correct entry
(c) A1 two correct entries of AB
(c) A1 AB = [[2,9],[-8,19]] cao
(c) Answer: AB = [[2,9],[-8,19]]
(d) M1 attempt at matrix multiplication BA with at least one correct entry
(d) A1 BA = [[15,20],[-1,6]] cao
(d) B1 correct conclusion that AB is not equal to BA (ft their AB, BA), so matrix multiplication is not commutative here oe
(d) Answer: BA = [[15,20],[-1,6]]; since AB = [[2,9],[-8,19]] is not equal to BA, multiplication is not commutative.

Question 2

(a) B1 det(M) = 2 cao
(a) Answer: det(M) = 2
(b) M1 correct adjugate [[4,-2],[-5,3]] with 1/det(M) used, ft their det(M)
(b) A1 M^-1 = [[2,-1],[-2.5,1.5]] oe cao
(b) Answer: M^-1 = (1/2)[[4,-2],[-5,3]] = [[2,-1],[-2.5,1.5]]
(c) M1 writing the system as M[x,y]^T = [8,14]^T and using [x,y]^T = M^-1[8,14]^T (ft their M^-1)
(c) A1 x = 2 cao
(c) A1 y = 1 cao
(c) Answer: x = 2, y = 1

Question 3

(a) B1 R = [[0,1],[1,0]] cao
(a) Answer: R = [[0,1],[1,0]]
(b) B1 S = [[0,-1],[1,0]] cao
(b) Answer: S = [[0,-1],[1,0]]
(c) M1 attempt at product RS with at least one correct entry, ft their R, S
(c) A1 two correct entries
(c) A1 RS = [[1,0],[0,-1]] cao
(c) Answer: RS = [[1,0],[0,-1]]
(d) B1 reflection in the x-axis (ft their RS if of the form [[1,0],[0,-1]])
(d) B1 image (3,2) cao
(d) Answer: RS represents a reflection in the x-axis; image of (3,-2) is (3,2).

Question 4

(a) M1 taking a general point (x, 2x) on the line and forming M(x,2x)^T
(a) M1 correctly computing both components: 5x - 2(2x) = x and 4x - 1(2x) = 2x
(a) A1 cso: image (x,2x) equals original point, so every point on y=2x is invariant
(a) Answer: Shown: M(x,2x)^T = (x,2x)^T for all x, so y = 2x is a line of invariant points.
(b) M1 considering a general point (x,mx) on line y=mx and forming M(x,mx)^T = (x(5-2m), x(4-m))
(b) M1 setting 4-m = m(5-2m) to require the image to lie on y=mx
(b) M1 forming and solving the quadratic 2m² - 6m + 4 = 0 oe, i.e. m²-3m+2=0
(b) A1 m = 2 (already found in (a)) or m = 1; other invariant line is y = x cao
(b) Answer: y = x
(c) B1 scale factor 3 (ft their line if consistent), oe
(c) Answer: Scale factor 3
(d) M1 M(1,1)^T = (3,3) found correctly (ft their M)
(d) A1 (3,3) lies on y=x, and (3,3) = 3*(1,1), confirming scale factor 3 cao
(d) Answer: M(1,1) = (3,3), which lies on y=x, confirming scale factor 3.

Question 5

(a) M1 correct expansion along a row or column: k(k²-1) - 1(k-1) + 1(1-k) oe
(a) A1 each of the three 2x2 minors correct
(a) M1 correct simplification of the expansion
(a) A1 cso: det(P) = k³ - 3k + 2 (answer printed)
(a) Answer: det(P) = k³ - 3k + 2
(b) M1 identifying k=1 as a root and factorising out (k-1), or equivalent factorisation attempt
(b) A1 cso: (k-1)²(k+2) confirmed by expansion matching k³-3k+2
(b) Answer: det(P) = (k-1)²(k+2)
(c) M1 setting (k-1)²(k+2) = 0 (ft their factorisation)
(c) A1 k = 1 or k = -2 cao, both values stated
(c) Answer: k = 1 or k = -2

Question 6

(a) B1 Q = [[2,3,1],[1,2,2],[3,1,1]] and r = (34,24,32)^T both correct oe
(a) Answer: Q = [[2,3,1],[1,2,2],[3,1,1]], r = (34,24,32)^T
(b) M1 correct expansion along a row or column, ft their Q
(b) A1 correct simplification
(b) A1 det(Q) = 10 cao
(b) Answer: det(Q) = 10
(c) M1 attempt at cofactor matrix of Q, ft their Q, det(Q)
(c) M1 correct adjugate (transpose of cofactor matrix)
(c) A1 Q^-1 = (1/10)[[0,-2,4],[5,-1,-3],[-5,7,1]] with at least 6 entries correct
(c) A1 fully correct Q^-1 cao
(c) Answer: Q^-1 = (1/10)[[0,-2,4],[5,-1,-3],[-5,7,1]]
(d) M1 computing v = Q^-1*r, ft their Q^-1
(d) A1 a = 8, c = 5 both correct
(d) A1 s = 3 cao, all three values stated with correct units (pounds)
(d) Answer: Adult ticket = 8 pounds, child ticket = 5 pounds, senior ticket = 3 pounds

Question 7

(a) B1 det(T) = 5 cao
(a) Answer: det(T) = 5
(b) B1 area scale factor = |det(T)| = 5, ft their det(T)
(b) B1 area of A'B'C' = 20 cao (using area ABC = 4)
(b) Answer: Area scale factor = 5; area of A'B'C' = 20
(c) M1 attempt at T applied to at least one vertex, ft their T
(c) A1 A' = (0,0) and B' = (12,4) both correct
(c) A1 C' = (2,4) cao
(c) Answer: A' = (0,0), B' = (12,4), C' = (2,4)
(d) M1 correct use of the shoelace (or equivalent) area formula with their coordinates
(d) A1 correct substitution: 0.5|0(4-4)+12(4-0)+2(0-4)|
(d) A1 area = 20 cao, matching part (b)
(d) Answer: Area = 20, confirming part (b)

Question 8

(a) M1 recognising k = √entries squared and summed, e.g. k = √1²+1²
(a) A1 k = √2
(a) M1 using cos(θ) = 1/√2 or sin(θ) = 1/√2 to find θ
(a) A1 cso: θ = 45 degrees, with √2*[[cos45,-sin45],[sin45,cos45]] = [[1,-1],[1,1]] confirmed
(a) Answer: k = √2, θ = 45 degrees
(b) B1 enlargement, scale factor √2, centre O (ft their k)
(b) B1 combined with rotation of 45 degrees anticlockwise about O (ft their θ)
(b) Answer: V represents an enlargement of scale factor √2 centre O, combined with a rotation of 45 degrees anticlockwise about O.
(c) M1 attempt at V*V with at least one correct entry
(c) A1 V² = [[0,-2],[2,0]] cao
(c) B1 enlargement scale factor 2, ft their V²
(c) B1 combined with rotation of 90 degrees anticlockwise about O, ft their V²
(c) Answer: V² = [[0,-2],[2,0]]; represents enlargement scale factor 2 combined with rotation 90 degrees anticlockwise about O.
(d) M1 attempt at V² applied to (3,1), ft their V²
(d) A1 (-2,6) cao
(d) Answer: (-2,6)

Question 9

(a) M1 correct expansion of det(A(λ)) along a row or column
(a) A1 correct simplification of all three terms
(a) A1 cso: det(A(λ)) = -3*λ (answer printed)
(a) Answer: det(A(λ)) = -3*λ
(b) B1 λ = 0, ft their part (a)
(b) Answer: λ = 0
(c) M1 adding equation 1 and equation 2 to eliminate z: 3x + 3y = 8
(c) M1 comparing with equation 3 at λ = 0: 3x + 3y = μ
(c) A1 μ = 8 cao
(c) Answer: μ = 8
(d) B1 (i) the three planes meet in a common line (a sheaf of planes), giving infinitely many solutions
(d) B1 (ii) the planes have no common point
(d) B1 (ii) they form a triangular prism, each pair of planes meeting in a line, with the three lines parallel and distinct oe
(d) Answer: (i) μ = 8: planes meet in a common line, infinitely many solutions. (ii) μ ≠ 8: planes form a triangular prism, no common point, system inconsistent.

Question 10

(a) M1 attempt at A*A with at least one correct entry
(a) A1 two correct entries
(a) A1 A² = [[18,7],[14,11]] cao
(a) Answer: A² = [[18,7],[14,11]]
(b) M1 finding trace(A)=7 and det(A)=10 and forming λ²-7lambda+10=0 (Cayley-Hamilton), or direct substitution using part (a)
(b) A1 cso: 7A - 10I = [[18,7],[14,11]], matching A² from part (a)
(b) Answer: A² = 7A - 10I confirmed
(c) M1 multiplying both sides of A²=7A-10I by A^-1 to obtain A = 7I - 10A^-1
(c) M1 rearranging to 10A^-1 = 7I - A
(c) A1 cso: A^-1 = 0.7I - 0.1A (answer printed)
(c) Answer: A^-1 = 0.7I - 0.1A
(d) B1 A^-1 = [[0.3,-0.1],[-0.2,0.4]] oe cao, ft their part (c)
(d) Answer: A^-1 = [[0.3,-0.1],[-0.2,0.4]] = (1/10)[[3,-1],[-2,4]]

Question 11

(a) M1 forming det(W-λ*I) = (7-λ)(-2-λ) - (-6)(3)
(a) A1 correct quadratic λ² - 5*λ + 4 = 0 oe
(a) M1 correct factorisation (λ-1)(λ-4) = 0
(a) A1 λ = 1, λ = 4, both cao
(a) Answer: λ = 1 or λ = 4
(b) M1 forming the image of (1,m) under W: (7-6m, 3-2m)
(b) M1 requiring the image to lie on y=mx: 3-2m = m(7-6m)
(b) M1 forming the quadratic 6m² - 9m + 3 = 0 oe, i.e. 2m²-3m+1=0
(b) M1 correct factorisation (2m-1)(m-1) = 0
(b) A1 both invariant lines stated: y = x and y = (1/2)x cao
(b) Answer: y = x and y = x/2
(c) B1 y = x, ft their lines
(c) B1 correct justification: this line corresponds to eigenvalue 1, so every point on it maps to itself (scale factor 1), e.g. W(1,1)=(1,1)
(c) Answer: y = x consists entirely of invariant points, since it corresponds to eigenvalue λ = 1.