# Row of a Matrix

## Definition

A straight horizontal path in a matrix to arrange the elements in the same path is called a row of a matrix.

The space inside the matrix is divided into number of rows according to following two factors.

1. The total number of elements.
2. The arrangement of number of elements equally in each row.

In every row, the elements are arranged in horizontally straight path but separated by some space.

It can be expressed in general form matrix form as follows.

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

The total number elements is $mn$. So, the space inside the matrix is divided into $m$ rows to place $n$ elements in each row.

Observe each element closely. Each element displays two numbers in its subscript position. The first number is common to each element in every row because it represents the number of that row.

For example, the number in subscript position of each element in first row is $1$ and it is common to all the elements in that row. The number $1$ represents the element belongs to first row and this rule is same for all the rows.

The general form matrix is simply written in compact form.

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

The element $e_{\displaystyle ij}$ represents every element in the matrix and the first letter $i$ in subscript position of the element represents the number of the respective element’s row.

### Example

Observe the following example matrices to understand the concept of row in matrix much better.

$\begin{bmatrix} \begin{array}{|c|} \hline 6 \\\hline \end{array} \end{bmatrix}$

There is only one element in this matrix. So, it is placed in row and therefore the element $e_{11} = 6$.

$\begin{bmatrix} \begin{array}{|cccc|} \hline 0 & 2 & 4 & 7 \\\hline \end{array} \end{bmatrix}$

The matrix has only one row but contains four elements. So, $e_{11} = 0$, $e_{12} = 2$, $e_{13} = 4$ and $e_{14} = 7$.

$\begin{bmatrix} \begin{array}{|cc|} \hline -1 & 9 \\\hline 5 & 8 \\\hline 2 & -5 \\\hline \end{array} \end{bmatrix}$

This matrix has three rows and two elements are placed in each row. Therefore, $e_{11} = -1$, $e_{12} = 9$, $e_{21} = 5$, $e_{22} = 8$, $e_{31} = 2$ and $e_{32} = -5$.

$\begin{bmatrix} \begin{array}{|cccc|} \hline 3 & 8 & -5 & 2 \\\hline 20 & -9 & -1 & -5 \\\hline 6 & 13 & 7 & 0 \\\hline 5 & -2 & -6 & 3 \\\hline \end{array} \end{bmatrix}$

This matrix has four rows and four elements are placed in each row. So, $e_{11} = 3$, $e_{12} = 8$, $e_{13} = -5$, $e_{14} = 2$, $e_{21} = 20$, $e_{22} = -9$, $e_{23} = -1$, $e_{24} = -5$, $e_{31} = 6$, $e_{32} = 13$, $e_{33} = 7$, $e_{34} = 0$, $e_{41} = 5$, $e_{42} = -2$, $e_{43} = -6$ and $e_{44} = 3$.