# Elements of a Matrix

## Definition

A quantity that displayed inside a matrix in a specific row and column is called an element of the matrix.

The space inside the matrix is split into number of rows and number of columns but every element is displayed inside every intersection area of row and column of the matrix. The elements are usually real numbers, literal numbers, combination of both, complex numbers or combination of all types of numbers.

### General form

An element can be represented in general form by a literal number algebraically but the representation of an element should represent the respective row and column to say the element belongs to the respective row and column. Due to this reason, the number of the row and number of the column are displayed as the subscript of the element.

$\begin{bmatrix} e_{11} & e_{12} & e_{13} & \cdots & e_{1n} \\ e_{21} & e_{22} & e_{23} & \cdots & e_{2n} \\ e_{31} & e_{32} & e_{33} & \cdots & e_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ e_{m1} & e_{m2} & e_{m3} & \cdots & e_{mn} \\ \end{bmatrix}$

Here, the literal $e$ is used to represent an element but two types of numbers are displayed at subscript position of the element. The first number in the subscript position of an element represent the number of the row and the second number in the subscript position of the same number denotes the number of the column.

For example

1. The element $e_{11}$ represent an element belongs to first row and first column.
2. The element $e_{21}$ expresses an element belongs to second row and first column.
3. The element $e_{45}$ denotes an element belongs to fourth row and fifth column.

#### Compact form

The general form matrix can be displayed in compact form by displaying only one element inside matrix but its row and column are expressed by two literals $i$ and $j$ and they are called $i^{\, th}$ row and $j^{\, th}$ column. Where $i$ and $j$ are natural numbers.

$\begin{bmatrix} e_{ij} \end{bmatrix}$

The element $e_{ij}$ represents every element in the matrix generally and $i = 1, 2, 3 \cdots$ and $j = 1, 2, 3 \cdots$

##### Examples

The following elements are some basic examples to understand how to display elements in rows and columns in matrices.

###### Example: 1

An example matrix having numbers as its elements.

$\begin{bmatrix} 3 & 6 & -7 \\ 8 & 0 & 4 \\ \end{bmatrix}$

In this matrix, $e_{11} = 3$, $e_{12} = 6$, $e_{13} = -7$, $e_{21} = 8$, $e_{22} = 0$ and $e_{23} = 4$

###### Example: 2

An example matrix, formed by the literal numbers.

$\begin{bmatrix} s & a^3t & u \\ f & w & g \\ u^2 & xyz & mn \\ \end{bmatrix}$

In this matrix, $e_{11} = s$, $e_{12} = a^3t$, $e_{13} = u$, $e_{21} = f$, $e_{22} = w$, $e_{23} = g$, $e_{31} = u^2$, $e_{32} = xyz$ and $e_{33} = mn$.

###### Example: 3

An example matrix with complex numbers

$\begin{bmatrix} 2 & 3i & 1-5i & -6i \\ 2+8i & 0 & 4 & 7 \\ \end{bmatrix}$

$e_{11} = 2$, $e_{12} = 3i$, $e_{13} = 1-5i$, $e_{14} = -6i$, $e_{21} = 2+8i$, $e_{22} = 0$, $e_{23} = 4$ and $e_{24} = 7$ in this matrix.