INTRODUCT|ON TO MATRICES

1. Introduction to Matrices

A matrix (plural: matrices) is a rectangular arrangement of numbers in rows and columns. Matrices help us organize and manipulate data efficiently.

In mathematics, we often need to organize numbers in an orderly way. For example, when recording marks of students, data in science experiments, or coordinates in physics, writing long lists of numbers can be confusing.

2. Definition of a Matrix

A matrix is a rectangular array of numbers arranged in rows (horizontal) and columns (vertical).

Illustration of a Matrix

$A = \begin{bmatrix} 2 & 4 & 6 \\ 1 & 3 & 5 \end{bmatrix}$

This matrix has:

2 rows
3 columns

So, it is called a 2 × 3 matrix (read as “two by three matrix”).

3. Elements (Entries) of a Matrix

Each number inside a matrix is called an element or entry.

For matrix $A$ A: $A = \begin{bmatrix} 2 & 4 & 6 \\ 1 & 3 & 5 \end{bmatrix}$

The element in the first row and second column is 4
We write it as $a_{12} = 4$

In general, the element in the i-th row and j-th column is written as $a_{ij}$

4. Types of Matrices

(a) Row Matrix

A matrix with only one row. $R = \begin{bmatrix} 3 & 5 & 7 \end{bmatrix}$

(b) Column Matrix

A matrix with only one column. $C = \begin{bmatrix} 2 \\ 4 \\ 6 \end{bmatrix}$

(c) Square Matrix

A matrix with the same number of rows and columns. $S = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$

This is a 2 × 2 square matrix.

(d) Zero (Null) Matrix

A matrix where all elements are zero. $Z = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$

5. Real-Life Example of Matrices

Suppose marks of three students in Mathematics and Physics are recorded as follows:

Student	Math	Physics
A	60	70
B	75	80
C	90	85

This can be written as a matrix: $M = \begin{bmatrix} 60 & 70 \\ 75 & 80 \\ 90 & 85 \end{bmatrix}$ M=607590708085

This matrix has 3 rows and 2 columns.

6. Basic Matrix Notation

A matrix is usually named by a capital letter, such as A, B, C, etc.

General form of a matrix: $A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}$

7. Simple Worked Examples on Matrices

Example 1

Write the order of the matrix: $B = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$

Solution:
Rows = 2, Columns = 3
Order = 2 × 3

Example 2

Find the element $a_{21}$ in the matrix: $A = \begin{bmatrix} 4 & 7 \\ 9 & 2 \end{bmatrix}$

Solution:
$a_{21}$ is in the 2nd row and 1st column = 9

8. Conclusion

Matrices are important in mathematics, science, engineering, computer graphics, economics, and many other fields. They help us store, organize, and perform operations on data in an efficient way. In later lessons, you will learn matrix addition, subtraction, multiplication, determinants, and applications.

PRACTICE QUESTIONS ON MATRICES

Section A: Basic Concepts

Question 1

State the order of the matrix: $A = \begin{bmatrix} 2 & 5 & 7 \\ 1 & 3 & 4 \end{bmatrix}$

Answer:
Rows = 2, Columns = 3
Order = 2 × 3

Question 2

Given the matrix: $B = \begin{bmatrix} 6 & 8 \\ 3 & 9 \end{bmatrix}$

Find:
a) $b_{11}$
b) $b_{12}$
c) $b_{21}$
d) $b_{22}$

Answer:
a) $b_{11} = 6$
b) $b_{12} = 8$
c) $b_{21} = 3$
d) $b_{22} = 9$

Question 3

Write a row matrix and a column matrix of your own.

Sample Answer:
Row matrix: $R = \begin{bmatrix} 4 & 6 & 8 \end{bmatrix}$

Column matrix: $C = \begin{bmatrix} 2 \\ 5 \\ 7 \end{bmatrix}$

Section B: Identifying Types of Matrices

Question 4

Classify the following matrices:

a) $A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

b) $B = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$

c) $C = \begin{bmatrix} 5 & 7 & 9 \end{bmatrix}$

Answer:
a) Square matrix
b) Zero (Null) matrix
c) Row matrix

Real-Life Application of Matrices

Question 5

The marks of two students in three subjects are recorded as follows:

Student	Math	English	Physics
John	65	70	80
Mary	75	60	85

a) Write the information in matrix form.
b) State the order of the matrix.

Answer:

a) $M = \begin{bmatrix} 65 & 70 & 80 \\ 75 & 60 & 85 \end{bmatrix}$ M=[657570608085]

b) Order = 2 × 3

Section D: Mixed Questions

Question 6

Given: $A = \begin{bmatrix} 3 & 4 & 5 \\ 6 & 7 & 8 \\ 1 & 2 & 9 \end{bmatrix}$

Find:
a) $a_{13}$
b) $a_{32}$
c) $a_{21}$

Answer:
a) $a_{13} = 5$
b) $a_{32} = 2$
c) $a_{21} = 6$

Question 7 (Introduction to Matrices)

Write a 2 × 2 zero matrix.

Answer: $Z = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$

Question 8 (Introduction to Matrices)

How many elements are in a 4 × 3 matrix?

Answer:
Number of elements = 4 × 3 = 12

Question 9

If matrix A has order 3 × 2, how many rows and columns does it have?

Answer:
Rows = 3
Columns = 2

Question 10

Create a square matrix of order 3 × 3 and label all its elements using $a_{ij}$ notation.

Sample Answer: $A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}$