Before you start

No specific prerequisites - this is a good place to start.

Method

Check the order (rows by columns) of each matrix before starting: two matrices can only be added or subtracted if they have exactly the same order.
To add or subtract matrices, combine the corresponding entries in the same position.
Before multiplying, check the two matrices are compatible: the number of columns in the first matrix must equal the number of rows in the second.
The resulting product matrix has as many rows as the first matrix and as many columns as the second.
Find each entry of the product by multiplying each row of the first matrix by the corresponding column of the second matrix, entry by entry, and summing the results.
In context questions, such as sales and prices, interpret each entry of the resulting matrix in terms of the real-world quantities it represents.

Worked example

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

Check compatibility: A is 2 x 2 and B is 2 x 2, so AB is defined and will be 2 x 2.
Find the (1,1) entry: (3)(2) + (-2)(-1) = 6 + 2 = 8.
Find the (1,2) entry: (3)(0) + (-2)(5) = 0 - 10 = -10.
Find the (2,1) entry: (1)(2) + (4)(-1) = 2 - 4 = -2.
Find the (2,2) entry: (1)(0) + (4)(5) = 0 + 20 = 20.
So AB = [[8, -10], [-2, 20]].

Practice questions

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Q1State the order of the matrix [[7, 2], [1, 8], [-3, 4]].Show answer

Answer: 3 x 2

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Q2A = [[3, -1], [2, 5]] and B = [[-2, 4], [1, -3]]. Find A + B.Show answer

Answer: [[1, 3], [3, 2]]

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Q3C = [[6, -2], [0, 3]]. Find 3C.Show answer

Answer: [[18, -6], [0, 9]]

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Q4D = [[4, 1], [-2, 3]] and E = [[1, -5], [0, 2]]. Find D - E.Show answer

Answer: [[3, 6], [-2, 1]]

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Q5F = [[2, -1], [0, 3]] and v = [[4], [-2]]. Find Fv.Show answer

Answer: [[10], [-6]]

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Q6G = [[2, -3], [1, 4]] and H = [[5, 1], [-2, 0]]. Find GH.Show answer

Answer: [[16, 2], [-3, 1]]

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Exam-style questions

Written in the style of a GCSE Further Maths exam paper, with a full mark scheme.

Q1[6 marks]

Given that [[a, 3b], [2c, -5]] + [[4, -7], [6, 1]] = [[9, -1], [16, -4]], find the values of a, b and c.

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Tick each line you got. Your score builds from the marks on the scheme.

forms the equation a + 4 = 9 from the (1,1) entries1 mark
solves to give a = 51 mark
forms the equation 3b - 7 = -1 from the (1,2) entries1 mark
solves to give b = 21 mark
forms the equation 2c + 6 = 16 from the (2,1) entries1 mark
solves to give c = 5; final answer: a = 5, b = 2, c = 51 mark

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Q2[3 marks]

P = [[2, -3], [4, 1]] and I is the 2 x 2 identity matrix. Show that PI = P.

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Tick each line you got. Your score builds from the marks on the scheme.

correct method for row 1 of PI: 2(1) + (-3)(0) = 2 and 2(0) + (-3)(1) = -31 mark
correct method for row 2 of PI: 4(1) + 1(0) = 4 and 4(0) + 1(1) = 11 mark
concludes PI = [[2, -3], [4, 1]], identical to P, as required; final answer: PI = [[2, -3], [4, 1]] = P (shown)1 mark

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Q3[6 marks]

A market stall sells umbrellas, hats and scarves. On Saturday it sold 8 umbrellas, 15 hats and 20 scarves; on Sunday it sold 11 umbrellas, 9 hats and 14 scarves. The prices, in pounds, are 9.50 for an umbrella, 6.00 for a hat and 4.50 for a scarf. (a) Write the Saturday and Sunday sales as a single combined row matrix. (b) Hence use matrix multiplication to calculate the total revenue for the weekend.

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adds corresponding entries: 8 + 11 = 19, 15 + 9 = 241 mark
20 + 14 = 34, giving the combined sales matrix [19, 24, 34]1 mark
sets up the multiplication 19(9.50) + 24(6.00) + 34(4.50)1 mark
19(9.50) = 180.501 mark
24(6.00) + 34(4.50) = 144 + 153 = 2971 mark
total revenue = 180.50 + 297 = 477.50; final answer: 477.50 pounds1 mark

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See real GCSE Further Maths past-paper questions, with official mark schemes →

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Matrix Operations: Addition, Subtraction and Multiplication

Before you start

Method

Worked example

Practice questions

Exam-style questions

Free printable worksheet

Next topics

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