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GCSE · Matrices

M1 Matrix Operations: Addition, Subtraction and Multiplication

AQA 8365 · Calculator allowed · about 105 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Four matrices are given below.
W = [[2, 1, 0], [3, -1, 4]]
X = [[5], [2], [-1]]
Y = [[1, 2], [3, 4], [0, -2]]
Z = [[6, 7]]
Which matrix has order 3 x 2 (that is, 3 rows and 2 columns)?

A) W
B) X
C) Y
D) Z

(Total for Question 1 is 1 mark)

P = [[3, -1, 2], [0, 5, 4]] and Q = [[7], [2], [-3]].

(a)State the order of matrix P.(1)

(b)State the order of matrix Q.(1)

(c)Explain whether the product PQ can be calculated. If it can, state the order of PQ.(2)

(Total for Question 2 is 4 marks)

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

(a)Find A + B.(2)

(b)Find A - B.(2)

(c)Write down B - A, using your answer to part (b) without further calculation.(1)

(Total for Question 3 is 5 marks)

D = [[4, -1], [0, 3]] and E = [[-2, 5], [1, -4]].

(a)Find 2D.(1)

(b)Find 3E.(1)

(c)Hence find 2D - 3E.(2)

(Total for Question 4 is 4 marks)

Given that [[x, 2y], [3z, -4]] + [[-3, 7], [9, 2]] = [[5, 15], [21, -2]], find the values of x, y and z.

(a)Find the value of x.(2)

(b)Find the value of y.(2)

(c)Find the value of z.(2)

(Total for Question 5 is 6 marks)

F = [[2, -3], [4, 1]] and v = [[5], [-2]]. Calculate the matrix Fv.

(Total for Question 6 is 2 marks)

G = [[1, 2], [3, -1]] and H = [[4, 0], [-2, 5]].

(a)Find GH.(3)

(b)Find HG.(3)

(c)Hence state, giving a reason, whether GH = HG.(1)

(Total for Question 7 is 7 marks)

A = [[2, -1], [3, 0]], B = [[1, 4], [-2, 5]] and C = [[7, 1], [7, 5]]. Given that kA + B = C, find the value of k.

(Total for Question 8 is 3 marks)

J = [[2, 0, -1], [3, 4, 1]] and K = [[1, 2], [0, -3], [5, 1]].

(a)State the order of the matrix JK.(1)

(b)Calculate JK.(4)

(Total for Question 9 is 5 marks)

A market stall sells three types of scarf: plain, striped and patterned. The number sold by two stallholders, Priya and Tom, is given by Q = [[12, 5, 8], [7, 10, 3]], where the rows represent Priya and Tom, and the columns represent plain, striped and patterned scarves. The prices in pounds sterling are given by the column matrix P = [[1.20], [1.50], [0.90]].

(a)Explain why the calculation QP can be carried out but PQ cannot.(2)

(b)Calculate QP, showing your method.(4)

(c)Interpret the two entries of QP in the context of the stallholders' sales.(2)

(Total for Question 10 is 8 marks)

A = [[a, 2], [1, b]]. Given that A x [[1], [-1]] = [[4], [-3]], find the values of a and b.

(a)Find the value of a.(2)

(b)Find the value of b.(2)

(Total for Question 11 is 4 marks)

A = [[2, -1], [3, 0]]. Find A² (that is, A x A).

(Total for Question 12 is 3 marks)

M = [[3, -2], [5, 4]] and I = [[1, 0], [0, 1]]. Show that MI = M.

(Total for Question 13 is 3 marks)

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

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

(b)Hence find (2A - B)C.(3)

(Total for Question 14 is 5 marks)

A = [[1, 0], [2, -1]], B = [[3, -2], [0, 1]] and C = [[1, 1], [2, 0]]. Show that (A + B)C = AC + BC for these matrices.

(a)Find (A + B)C.(3)

(b)Find AC + BC, and hence state whether (A + B)C = AC + BC for these matrices.(4)

(Total for Question 15 is 7 marks)

A = [[2, 4], [1, 2]] and B = [[x, -8], [-1, y]]. Given that AB is the zero matrix [[0, 0], [0, 0]], find the values of x and y.

(a)Find the value of x.(2)

(b)Find the value of y.(2)

(c)State what this result shows about matrix multiplication, compared with multiplication of real numbers.(1)

(Total for Question 16 is 5 marks)

A = [[1, 2], [0, 1]] and B = [[5, 4], [1, 3]]. X is a 2 x 2 matrix such that AX = B.

(a)Let X = [[p, q], [r, s]]. Show that AX = [[p + 2r, q + 2s], [r, s]].(2)

(b)Hence find the matrix X.(4)

(Total for Question 17 is 6 marks)

A cafe records the number of coffees, teas and cakes sold in a row matrix each week. In Week 1 the sales were [50, 35, 15] and in Week 2 the sales were [42, 28, 24] (coffees, teas, cakes respectively). The prices, in pounds sterling, are given by the column matrix [[2.50], [2.00], [3.20]].

(a)Find the total sales matrix for the two weeks combined (Week 1 + Week 2).(2)

(b)Hence calculate the total revenue for the two weeks combined, using matrix multiplication.(3)

(c)The cafe owner plans to increase cake sales by 20% next week, while coffee and tea sales stay the same as the Week 2 total. Write down the new sales row matrix and find the new predicted revenue.(4)

(Total for Question 18 is 9 marks)

Mark scheme · M1 Matrix Operations: Addition, Subtraction and Multiplication

Question 1

B1 C cao
Answer: C) Y

Question 2

(a) B1 2 x 3 cao
(a) Answer: 2 x 3
(b) B1 3 x 1 cao
(b) Answer: 3 x 1
(c) B1 valid reason: PQ can be calculated because the number of columns of P (3) equals the number of rows of Q (3) oe
(c) B1 states the order of PQ is 2 x 1 (rows of P by columns of Q)
(c) Answer: Yes, PQ can be calculated; PQ has order 2 x 1

Question 3

(a) M1 adds corresponding entries, with at least two entries correct
(a) A1 [[3, -2], [8, -1]] cao
(a) Answer: [[3, -2], [8, -1]]
(b) M1 subtracts corresponding entries, with at least two entries correct
(b) A1 [[7, -4], [-4, 9]] cao
(b) Answer: [[7, -4], [-4, 9]]
(c) B1 [[-7, 4], [4, -9]], the negative of the matrix found in (b)
(c) Answer: [[-7, 4], [4, -9]]

Question 4

(a) B1 [[8, -2], [0, 6]] cao
(a) Answer: [[8, -2], [0, 6]]
(b) B1 [[-6, 15], [3, -12]] cao
(b) Answer: [[-6, 15], [3, -12]]
(c) M1 subtracts corresponding entries of 3E from 2D using answers to (a) and (b)
(c) A1 [[14, -17], [-3, 18]] cao
(c) Answer: [[14, -17], [-3, 18]]

Question 5

(a) M1 forms the equation x + (-3) = 5 from the (1,1) entries oe
(a) A1 x = 8 cao
(a) Answer: x = 8
(b) M1 forms the equation 2y + 7 = 15 from the (1,2) entries oe
(b) A1 y = 4 cao
(b) Answer: y = 4
(c) M1 forms the equation 3z + 9 = 21 from the (2,1) entries oe
(c) A1 z = 4 cao
(c) Answer: z = 4

Question 6

M1 at least one correct entry: 2(5)+(-3)(-2)=16 or 4(5)+1(-2)=18
A1 [[16], [18]] cao
Answer: [[16], [18]]

Question 7

(a) M1 correct method for row 1 of GH: 1(4)+2(-2)=0 and 1(0)+2(5)=10
(a) M1 correct method for row 2 of GH: 3(4)+(-1)(-2)=14 and 3(0)+(-1)(5)=-5
(a) A1 [[0, 10], [14, -5]] cao
(a) Answer: [[0, 10], [14, -5]]
(b) M1 correct method for row 1 of HG: 4(1)+0(3)=4 and 4(2)+0(-1)=8
(b) M1 correct method for row 2 of HG: -2(1)+5(3)=13 and -2(2)+5(-1)=-9
(b) A1 [[4, 8], [13, -9]] cao
(b) Answer: [[4, 8], [13, -9]]
(c) B1 GH is not equal to HG, e.g. the (1,1) entries differ (0 compared with 4), so matrix multiplication is not commutative here
(c) Answer: GH is not equal to HG

Question 8

M1 forms an equation from one pair of entries, e.g. 2k + 1 = 7
A1 k = 3 cao
B1 checks k = 3 against a second entry, e.g. 3(3) - 2 = 7, confirming consistency
Answer: k = 3

Question 9

(a) B1 2 x 2 cao
(a) Answer: 2 x 2
(b) M1 correct method for row 1: 2(1)+0(0)+(-1)(5)=-3 and 2(2)+0(-3)+(-1)(1)=3
(b) M1 correct method for row 2: 3(1)+4(0)+1(5)=8 and 3(2)+4(-3)+1(1)=-5
(b) A1 first row correct: [-3, 3]
(b) A1 second row correct: [8, -5], giving JK = [[-3, 3], [8, -5]] cao
(b) Answer: [[-3, 3], [8, -5]]

Question 10

(a) B1 for QP: the order of Q is 2 x 3 and the order of P is 3 x 1; the number of columns of Q (3) equals the number of rows of P (3), so QP is defined oe
(a) B1 for PQ: the number of columns of P (1) does not equal the number of rows of Q (2), so PQ is not defined oe
(a) Answer: QP is defined because the inner dimensions match (3=3); PQ is not defined because they do not (1 is not equal to 2)
(b) M1 correct method for Priya's row: 12(1.20)+5(1.50)+8(0.90)
(b) M1 correct method for Tom's row: 7(1.20)+10(1.50)+3(0.90)
(b) A1 first entry correct: 29.10 (GBP)
(b) A1 second entry correct: 26.10 (GBP), giving QP = [[29.10], [26.10]] cao
(b) Answer: [[29.10], [26.10]] (in GBP)
(c) B1 states that GBP29.10 is the total revenue Priya earned from her scarf sales
(c) B1 states that GBP26.10 is the total revenue Tom earned from his scarf sales
(c) Answer: Priya's total revenue is GBP29.10 and Tom's total revenue is GBP26.10

Question 11

(a) M1 forms the equation a(1) + 2(-1) = 4, i.e. a - 2 = 4
(a) A1 a = 6 cao
(a) Answer: a = 6
(b) M1 forms the equation 1(1) + b(-1) = -3, i.e. 1 - b = -3
(b) A1 b = 4 cao
(b) Answer: b = 4

Question 12

M1 correct method for row 1: 2(2)+(-1)(3)=1 and 2(-1)+(-1)(0)=-2
M1 correct method for row 2: 3(2)+0(3)=6 and 3(-1)+0(0)=-3
A1 [[1, -2], [6, -3]] cao
Answer: [[1, -2], [6, -3]]

Question 13

M1 correct method for row 1 of MI: 3(1)+(-2)(0)=3 and 3(0)+(-2)(1)=-2
M1 correct method for row 2 of MI: 5(1)+4(0)=5 and 5(0)+4(1)=4
A1 concludes MI = [[3, -2], [5, 4]], which is identical to M, as required (answer printed)
Answer: MI = [[3, -2], [5, 4]] = M (shown)

Question 14

(a) M1 finds 2A = [[2,4],[0,-2]] and subtracts B correctly entry-wise
(a) A1 [[5, 3], [-2, -6]] cao
(a) Answer: [[5, 3], [-2, -6]]
(b) M1 correct method for row 1: 5(0)+3(1)=3 and 5(2)+3(-2)=4
(b) M1 correct method for row 2: -2(0)+(-6)(1)=-6 and -2(2)+(-6)(-2)=8
(b) A1 [[3, 4], [-6, 8]] cao
(b) Answer: [[3, 4], [-6, 8]]

Question 15

(a) M1 finds A + B = [[4, -2], [2, 0]]
(a) M1 correct method for multiplying (A+B) by C
(a) A1 [[0, 4], [2, 2]] cao
(a) Answer: [[0, 4], [2, 2]]
(b) M1 finds AC = [[1, 1], [0, 2]] correctly
(b) M1 finds BC = [[-1, 3], [2, 0]] correctly
(b) A1 AC + BC = [[0, 4], [2, 2]] cao
(b) B1 correct conclusion that AC + BC = [[0,4],[2,2]] is identical to (A+B)C found in (a), so (A+B)C = AC + BC is confirmed for these matrices
(b) Answer: AC + BC = [[0, 4], [2, 2]], which equals (A+B)C, confirming the result

Question 16

(a) M1 forms an equation from an entry of AB containing x, e.g. 2x + 4(-1) = 0
(a) A1 x = 2 cao
(a) Answer: x = 2
(b) M1 forms an equation from an entry of AB containing y, e.g. 2(-8) + 4y = 0
(b) A1 y = 4 cao
(b) Answer: y = 4
(c) B1 states that, unlike real numbers, two non-zero matrices can multiply together to give the zero matrix oe
(c) Answer: Two non-zero matrices can have a product equal to the zero matrix, which is impossible for two non-zero real numbers

Question 17

(a) M1 multiplies A by X using the general entries p, q, r, s
(a) A1 correctly shows AX = [[p+2r, q+2s], [r, s]] (answer printed)
(a) Answer: AX = [[p + 2r, q + 2s], [r, s]] (shown)
(b) M1 compares the bottom row to find r = 1 and s = 3 directly
(b) A1 r = 1 and s = 3 cao
(b) M1 substitutes r and s into the top row equations, p+2(1)=5 and q+2(3)=4, to find p and q
(b) A1 states X = [[3, -2], [1, 3]] cao
(b) Answer: X = [[3, -2], [1, 3]]

Question 18

(a) M1 adds corresponding entries of the two row matrices
(a) A1 [92, 63, 39] cao
(a) Answer: [92, 63, 39]
(b) M1 multiplies the combined sales matrix by the price matrix, forming 92(2.50)+63(2.00)+39(3.20)
(b) M1 correct method evaluating each product: 230 + 126 + 124.80
(b) A1 GBP480.80 cao
(b) Answer: GBP480.80
(c) M1 calculates the new cake figure as 24 x 1.20 = 28.8, keeping coffees at 42 and teas at 28
(c) A1 states the new sales matrix as [42, 28, 28.8]
(c) M1 multiplies the new sales matrix by the price matrix: 42(2.50)+28(2.00)+28.8(3.20)
(c) A1 GBP253.16 cao
(c) Answer: New sales matrix [42, 28, 28.8]; new predicted revenue GBP253.16

Matrix Operations: Addition, Subtraction and Multiplication - Worksheets, Questions and Revision