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What is a matrix?

A matrix (plural matrices) is a rectangular array of numbers, displayed in rows and columns inside a large set of brackets.

One use of matrices is to organise data clearly.

For example, the number of people that attended an exhibition over one weekend can be arranged in a matrix.

This is a 3 × 2 (“3 by 2”) matrix because it has 3 rows and 2 columns. It contains 6 elements or entries.

Men 129 105

Saturday Sunday

Women 103 99

Children 80 67

129 105

103 99

80 67

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Adding matrices

Two matrices can be added or subtracted if they have the same dimensions.

For example:

6 5

–2 13

8 –17

+

6 + 12 5 + 4

–2 + 1 13 – 3

8 + 2 –17 + 0

=

12 4

1 –3

2 0

Add each corresponding element from both matrices to get the resulting element.

=

18 9

–1 10

10 –17

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Adding and subtracting matrices

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Multiplying by a scalar

A matrix can be multiplied by a single value (a scalar).

Simply multiply each entry in the matrix by that scalar to get the resulting matrix.

For example:

Calculate:

3

7 5

3 2

11 1

=

7 × 3 5 × 3

3 × 3 2 × 3

11 × 3 1 × 3

=

21 15

9 6

33 3

53 2

11 1–

1 2

3 2

=15 10

55 5–

1 2

3 2=

14 8

52 3

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Multiplying two matrices

List all possible product pairs from the matrices below.

Two matrices A and B can be multiplied, but only if the number of columns in matrix A equals the number of rows in matrix B.

An m × n matrix can be multiplied by an n × p matrix, and the result is an m × p matrix.

1

4

7

12 15

13 9

12 7

5 7

3 3

129

103A = B = C = D =

Unlike with numbers, the order in which two matrices are multiplied does matter, i.e. AB ≠ BA as a rule.

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How to multiply two matrices

To multiply two matrices, perform the dot product on rows and columns of the matrices.

For larger matrices, start with the first row of the first matrix and perform the dot product on each column of the second.Work through each row of the first matrix in this way.

The dot product is the sum of the product of the corresponding entries.

For example: 4

5

6

1 2 3 = (1 × 4) + (2 × 5) + (3 × 6)

= 4 + 10 + 18 = 32

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Multiplying two matrices