Introduction to Matrices

You may not notice it, but matrices appear naturally in many everyday situations. For example:

• A marks table showing marks of students in different subjects.
• A seating plan of a classroom (rows of benches and columns of students).
• A table showing monthly sales for different products.

All these are actually matrices: numbers arranged in rows and columns.

Here is a simple example of a matrix with 2 rows and 3 columns:

We can also imagine it as a small table:

This is the basic picture you should keep in mind whenever you hear the word matrix.

A general matrix with m rows and n columns is written like this:

Here:

• \( m \) = number of rows
• \( n \) = number of columns
• \( a_{ij} \) = entry in the i-th row and j-th column

The symbol \( a_{ij} \) is read as a i j and it tells you exactly where the element is placed inside the matrix:

• The first index \( i \) tells you the row number.
• The second index \( j \) tells you the column number.

For example, if

then:

• \( a_{11} = 3 \) (row 1, column 1)
• \( a_{12} = -1 \) (row 1, column 2)
• \( a_{23} = 5 \) (row 2, column 3)

To find the order of a matrix:

1. Count the number of horizontal rows.
2. Count the number of vertical columns.
3. Write it as \( \text{(number of rows)} \times \text{(number of columns)} \).

For example,

Here, there are 2 rows and 3 columns, so the order of \( B \) is \( 2 \times 3 \).

Look at these matrices and their orders:

• \( \begin{bmatrix} 4 & 9 \end{bmatrix} \) has order \( 1 \times 2 \) (1 row, 2 columns).
• \( \begin{bmatrix} 2 \\ -3 \\ 5 \end{bmatrix} \) has order \( 3 \times 1 \) (3 rows, 1 column).
• \( \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \) has order \( 3 \times 3 \).

Two matrices can be easily compared using their orders. Later, you will see that many operations on matrices are possible only when their orders match certain rules.

Consider the matrix

Here:

• Row 1: \( [2, -1, 4] \)
• Row 2: \( [0, 3, 5] \)
• Row 3: \( [7, 1, -2] \)

and

• Column 1: \( [2, 0, 7] \)
• Column 2: \( [-1, 3, 1] \)
• Column 3: \( [4, 5, -2] \)

To describe the position of an element, we always use the pattern:

\( a_{ij} = \) element in the i-th row and j-th column.

In the matrix \( C \) above:

• \( c_{11} = 2 \)
• \( c_{13} = 4 \)
• \( c_{21} = 0 \)
• \( c_{32} = 1 \)

Whenever you see a symbol like \( a_{23} \), immediately think: “row 2, column 3”. This habit will make matrix problems much easier.

Suppose three students A, B and C have marks in Maths and Science as follows:

We can write this information as a matrix:

Now all the data is captured in one matrix \( M \). Later, we will learn how to use such matrices to perform operations like finding totals, averages and more.

Consider the system of equations:

We can write this system using matrices as:

This matrix form makes it easier to apply powerful methods to solve the system. You will study these methods in later topics.