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URL: https://nakafa.com/en/subjects/mathematics/matrix/matrix-concept
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Understand matrix fundamentals from scratch. Learn rows, columns, elements, and matrix order with real-world examples and clear visual explanations.

---

## 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 $$[]$$.

Visible text: 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        | $$95$$ | $$75$$ |

Visible text: | Student Name | Math Score | English Score |
| :----------- | :--------- | :------------ |
| Aisyah | | |
| Alex | | |
| Wayan | | |

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

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

Or it can also be written with parentheses:

```math
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 $$A$$.

Visible text: In this example, the numbers are the elements of matrix .

## Rows, Columns, and Matrix Elements

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

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

Visible text: 1. **Row**: An arrangement of elements horizontally.
2. **Column**: An arrangement of elements vertically.
3. **Matrix Element**: Each number or entry within the matrix.

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

Visible text: Let's look back at matrix from the previous example:

```math
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 $$\begin{bmatrix} 75 \\ 95 \\ 75 \end{bmatrix}$$

Visible text: - **Row** is 
- **Row** is 
- **Row** is 

- **Column** is 
- **Column** is

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.

Visible text: Matrix elements are usually denoted by a lowercase letter corresponding to the matrix name, with two indices, for example . The first index () indicates the row position, and the second index () indicates the column position.

So, for matrix $$A$$ above:

Visible text: So, for matrix 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 $$75$$.

Visible text: - is the element in the st row, st column, which is .
- is the element in the st row, nd column, which is .
- is the element in the nd row, st column, which is .
- is the element in the nd row, nd column, which is .
- is the element in the rd row, st column, which is .
- is the element in the rd row, nd column, which is .

## 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").

Visible text: 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 rows and columns, then the matrix is said to be of order (read as "m by n").

Consider our matrix $$A$$ again:

Visible text: Consider our matrix again:

```math
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 $$A_{3 \times 2}$$.

Visible text: Matrix has rows (indicated by the curly brace on the right) and columns (indicated by the curly brace below). So, the order of matrix is . We can write it as .

**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        | $$1$$ | $$2$$ | $$1$$ |

Visible text: | Student Name | Permitted | Sick | Unexcused |
| :----------- | :-------- | :--- | :-------- |
| Aisyah | | | |
| Alex | | | |
| Wayan | | | |

If we convert this into matrix $$B$$:

Visible text: If we convert this into matrix :

```math
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 $$B_{3 \times 3}$$.

Visible text: Matrix has rows (indicated by the curly brace on the right) and columns (indicated by the curly brace below). Thus, the order of matrix is . We can write it as .

Elements of matrix $$B$$, for example:

Visible text: Elements of matrix , 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 $$2$$)

Visible text: - (element row , column )
- (element row , column )
- (element row , column )

## General Notation of a Matrix

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

Visible text: In general, a matrix with `m` rows and `n` columns can be written as follows:

```math
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.

Visible text: - : Matrix of order 
 .
- : Number of rows.
- : Number of columns.
- : Element of matrix in the
 
 -th row and 
 -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!