Matrix Concept: Matrix - Mathematics (Grade 11)

What is a Matrix?

Have you ever seen a list of test scores arranged in a table? Or perhaps a team duty roster? Unconsciously, we often encounter data presented in rows and columns. Well, this arrangement of numbers or information in rows and columns is the basis of the matrix concept.

A matrix is a collection of numbers specially arranged in rows and columns to form a rectangular array. The numbers that make up this matrix are called matrix elements. Matrices are usually written within parentheses $()$ or square brackets $[]$ .

For example, consider the Math and English test score data for the following three students:

Student Name	Math Score	English Score
Aisyah	$80$	$75$
Alex	$70$	$95$
Wayan	$95$	$75$

The data above can be presented in matrix form. If we only take the numbers, the matrix will look like this:

A = \begin{bmatrix} 80 & 75 \\ 70 & 95 \\ 95 & 75 \end{bmatrix}

Or it can also be written with parentheses:

A = \begin{pmatrix} 80 & 75 \\ 70 & 95 \\ 95 & 75 \end{pmatrix}

In this example, the numbers $80, 75, 70, 95, 95, 75$ are the elements of matrix $A$ .

Rows, Columns, and Matrix Elements

In a matrix, there are several important terms you need to know:

Row: An arrangement of elements horizontally.
Column: An arrangement of elements vertically.
Matrix Element: Each number or entry within the matrix.

Let's look back at matrix $A$ from the previous example:

A = \begin{bmatrix} 80 & 75 \\ 70 & 95 \\ 95 & 75 \end{bmatrix}

Row $1$ is $[80 \ 75]$
Row $2$ is $[70 \ 95]$
Row $3$ is $[95 \ 75]$
Column $1$ is $\begin{bmatrix} 80 \\ 70 \\ 95 \end{bmatrix}$
Column $2$ is $\begin{bmatrix} 75 \\ 95 \\ 75 \end{bmatrix}$

Matrix elements are usually denoted by a lowercase letter corresponding to the matrix name, with two indices, for example $a_{ij}$ . The first index ( $i$ ) indicates the row position, and the second index ( $j$ ) indicates the column position.

So, for matrix $A$ above:

$a_{11}$ is the element in the $1$ st row, $1$ st column, which is $80$ .
$a_{12}$ is the element in the $1$ st row, $2$ nd column, which is $75$ .
$a_{21}$ is the element in the $2$ nd row, $1$ st column, which is $70$ .
$a_{22}$ is the element in the $2$ nd row, $2$ nd column, which is $95$ .
$a_{31}$ is the element in the $3$ rd row, $1$ st column, which is $95$ .
$a_{32}$ is the element in the $3$ rd row, $2$ nd column, which is $75$ .

Order of a Matrix

Every matrix has a size called its order. The order of a matrix is determined by the number of rows and columns it has. If a matrix has $m$ rows and $n$ columns, then the matrix is said to be of order $m \times n$ (read as "m by n").

Consider our matrix $A$ again:

A = \underbrace{\begin{bmatrix} 80 & 75 \\ 70 & 95 \\ 95 & 75 \end{bmatrix}}_{\text{2 columns}} \hspace{0.1em} \left. \vphantom{\begin{bmatrix} 80 \\ 70 \\ 95 \end{bmatrix}} \right\} \hspace{0.1em} \text{3 rows}

Matrix $A$ has $3$ rows (indicated by the curly brace on the right) and $2$ columns (indicated by the curly brace below). So, the order of matrix $A$ is $3 \times 2$ . We can write it as $A_{3 \times 2}$ .

Another Example:

Suppose we have student attendance data for one semester:

Student Name	Permitted	Sick	Unexcused
Aisyah	$2$	$1$	$0$
Alex	$3$	$1$	$1$
Wayan	$1$	$2$	$1$

If we convert this into matrix $B$ :

B = \underbrace{\begin{bmatrix} 2 & 1 & 0 \\ 3 & 1 & 1 \\ 1 & 2 & 1 \end{bmatrix}}_{\text{3 columns}} \hspace{0.1em} \left. \vphantom{\begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix}} \right\} \hspace{0.1em} \text{3 rows}

Matrix $B$ has $3$ rows (indicated by the curly brace on the right) and $3$ columns (indicated by the curly brace below). Thus, the order of matrix $B$ is $3 \times 3$ . We can write it as $B_{3 \times 3}$ .

Elements of matrix $B$ , for example:

$b_{11} = 2$ (element row $1$ , column $1$ )
$b_{23} = 1$ (element row $2$ , column $3$ )
$b_{32} = 2$ (element row $3$ , column $2$ )

General Notation of a Matrix

In general, a matrix $A$ with m rows and n columns can be written as follows:

A_{m \times n} = \underbrace{\begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \dots & a_{mn} \end{bmatrix}}_{\text{n columns}} \hspace{0.1em} \left. \vphantom{\begin{bmatrix} a_{11} \\ a_{21} \\ \vdots \\ a_{m1} \end{bmatrix}} \right\} \hspace{0.1em} \text{m rows}

Key:

$A_{m \times n}$ : Matrix $A$ of order $m \times n$ .
$m$ : Number of rows.
$n$ : Number of columns.
$a_{ij}$ : Element of matrix $A$ in the $i$ -th row and $j$ -th column.

By understanding this basic concept, you are ready to learn more about the types of matrices and the operations that can be performed on them. Matrices are very useful tools in various fields, you know, from mathematics, physics, computer science, to economics!