Matrices are arrays of numbers arranged in rows and columns. They are enclosed in brackets and referenced by letters. The order or size of a matrix is denoted by the number of rows and columns, such as a 3x4 matrix. There are several types of matrices including column matrices, row matrices, square matrices, and identity matrices. Matrices can be added, subtracted, or multiplied through scalar or element-wise operations depending on their size and type. Matrix multiplication involves multiplying the rows of the first matrix by the columns of the second.

1. MATRICES

2. Definition
A group of numbers (called elements) arranged in rows and
columns and enclosed in brackets.
4 columns
5 rows
Cricket scores of five batsmen against
opposing teams P, Q, R and S
𝐵 =
12 15
45 38
15 17
3 rows
2 columns

3. Definition - examples
The exam grades for five students in a
class
𝑮 =
70 72 80
78
89
90
81
85
92
88
67
99
88
86
78
3 columns
5 rows
Items 2019 2020 2021
Chicken 4 3 4
Sugar 1 2 2
Rice 2 3 2
Price of common items to the nearest
dollar
𝑃 =
4 3 4
1
2
2
3
2
2
3 columns
3 rows

4. Dimension or size of matrices
A matrix with m rows and n columns is called a matrix of
order m x n or a m x n matrix
3 −2 7 9
1
−8
0
2
−3
10
5
−6
2 8
1
6
0
3
20 7 54 6
11 12 76 80
13 −2 27 91
1
−18
17
32
−3
10
75
−6
4
11
3
31
23
10
2
73
0
85
43
2 11 7 9 −2
1 0
0 1
2 3 7
0 1 0
5 0 1
2 3 6 2
0 9 10 0
5 8 0 1

5. Types of Matrices
Column matrix – a
matrix with one
column.
Size: m x 1
𝐴 =
4
11
3
𝐵 =
15
5
10
9
Row matrix – a
matrix with one
row.
Size: 1 x m
𝐴 = 2 11 7 9 −2
𝐵 = 23 16 11
Square matrix – a
matrix with same
number of rows
and columns.
Size: m x m
𝐴 =
2 5
3 7
𝐵 =
0 4 13
1 7 10
0 3 8
Identity matrix –
a square matrix in
which the diagonal
elements are all 1
and the rest is
zero.
Size: m x m
𝐴 =
1 0
0 1
𝐵 =
1 0 0
0 1 0
0 0 1

6. Addition/Subtraction
of Matrices
• Only matrices of the
same size can be added
or subtracted
• Corresponding elements
must be added or
subtracted

7. Addition/Subtraction of Matrices

8. Addition/Subtraction of Matrices
𝑨 − 𝑩 =
𝟑 𝟏
𝟓 𝟒
−
𝟓 𝟎
−𝟐 𝟕
=
𝟑 − 𝟓 𝟏 − 𝟎
𝟓 − −𝟐 𝟒 − 𝟕
=
−𝟐 𝟏
𝟕 −𝟑
𝑩 − 𝑨 =
𝟓 𝟎
−𝟐 𝟕
−
𝟑 𝟏
𝟓 𝟒
=
𝟓 − 𝟑 𝟎 − 𝟏
−𝟐 − 𝟓 𝟕 − 𝟒
=
𝟐 −𝟏
−𝟕 𝟑
𝑨 + 𝑪 − 𝑩 =
𝟑 𝟏
𝟓 𝟒
+
−𝟏 𝟑
−𝟒 𝟎
−
𝟓 𝟎
−𝟐 𝟕
=
𝟑 − 𝟏 − 𝟓 𝟏 + 𝟑 − 𝟎
𝟓 − 𝟒 − −𝟐 𝟒 + 𝟎 − 𝟕
=
−𝟑 𝟒
𝟑 −𝟑
𝑪 − 𝑨 + 𝑩 =
−𝟏 𝟑
−𝟒 𝟎
−
𝟑 𝟏
𝟓 𝟒
+
𝟓 𝟎
−𝟐 𝟕
=
−𝟏 − 𝟑 + 𝟓 𝟑 − 𝟑 + 𝟎
−𝟒 − 𝟓 − 𝟐 𝟎 − 𝟒 + 𝟕
=
𝟏 𝟐
−𝟏𝟏 𝟑

9. Multiplication of Matrices – Scalar
A scalar is a constant or a single element or number. Scalar multiplication is done by
multiplying each element in the matrix by the scalar

10. Multiplication of Matrices – Scalar
3

11. Multiplication of Matrices by a Scalar

12. Multiplication
of Matrices
• Multiply each row of the
first matrix by the columns
of the second matrix
• The number of columns in
the first matrix must be the
same as the number of rows
in the second matrix
2 5
7 1
3 6
2 5
8 3

13. Multiplication of Matrices
10 35
8 17
21 −3
8 2
−7 7
5 27
23 9
−8 5

14. Multiplication of Matrices
3
24 52
29 50
46 −6
−1 15
2 3
4 2
6 18
12 12
𝑓 = 0
e = 1
𝑎 = 1
𝑏 = 4
𝑐 = 6
d = 2