Position of entries in a Matrix

Definition

A place where an entry is located in a matrix, is called the position of an entry in a matrix.

Introduction

The elements are displayed in a matrix. There is no issue if a matrix consists of an element but the position of an entry comes into consideration when a matrix consists of more than one element. So, it is essential for everyone to learn how to find the position of an element in a matrix.

position of an element or entry in a matrix

In a matrix, the entries are actually placed in the rows and columns, which means the position of an entry depends on both the number of the row and the number of the column. So, the location of every entry in a matrix should be recognized by its row and column.

Examples

Let’s see some examples to find the positions of all elements in matrices.

$\left[\begin{array}{rrr} 8 & -5 & 4 \\ -7 & 9 & -6 \\ \end{array}\right]$

Draw the horizontal lines above and below the entries in the matrix.

$\left[\begin{array}{rrr} \hline 8 & -5 & 4 \\ \hline -7 & 9 & -6 \\ \hline \end{array}\right]$

Now, the elements $8,$ $-5$ and $4$ belong to the first row. Similarly, the elements $-7,$ $9$ and $-6$ belong to the second row.

Similarly, draw a vertical line before and after the elements in the matrix.

$\left[\begin{array}{|r|r|r|} 8 & -5 & 4 \\ -7 & 9 & -6 \\ \end{array}\right]$

Now, the entries $8$ and $-7$ belong to the first column, the entries $-5$ and $9$ belong to the second column and the entries $4$ and $-6$ belong to the third column.

It is time to learn how to find the position of every element in the matrix.

$(1).\,\,$ $\left[\begin{array}{|r|rr} \hline 8 & -5 & 4 \\ \hline -7 & 9 & -6 \\ \end{array}\right]$

The entry $8$ belongs to the first row and also belongs to the first column. So, the position of an entry $8$ is the first row and first column.

$(2).\,\,$ $\left[\begin{array}{r|r|r} \hline 8 & -5 & 4 \\ \hline -7 & 9 & -6 \\ \end{array}\right]$

The entry $-5$ belongs to the first row and also belongs to the second column. So, the position of an entry $-5$ is the first row and second column.

$(3).\,\,$ $\left[\begin{array}{rr|r|} \hline 8 & -5 & 4 \\ \hline -7 & 9 & -6 \\ \end{array}\right]$

The entry $4$ belongs to the first row and also belongs to the third column. So, the position of an entry $4$ is the first row and third column.

$(4).\,\,$ $\left[\begin{array}{|r|rr} 8 & -5 & 4 \\ \hline -7 & 9 & -6 \\ \hline \end{array}\right]$

The entry $-7$ belongs to the second row and also belongs to the first column. So, the position of an entry $-7$ is the second row and first column.

$(5).\,\,$ $\left[\begin{array}{r|r|r} 8 & -5 & 4 \\ \hline -7 & 9 & -6 \\ \hline \end{array}\right]$

The entry $9$ belongs to the second row and also belongs to the second column. So, the position of an entry $9$ is the second row and second column.

$(6).\,\,$ $\left[\begin{array}{rr|r|} 8 & -5 & 4 \\ \hline -7 & 9 & -6 \\ \hline \end{array}\right]$

The entry $-6$ belongs to the second row and also belongs to the third column. So, the position of an entry $-6$ is the second row and third column.

In this way, the position of every element can be determined in the matrix.

Uses

The position of an entry in matrices is mainly useful in two cases.

It is primarily useful to identify the exact location of every entry in a matrix.
It is also used to express the location of any element in a matrix.

Representation

An element with its exact position is written as $e$ subscript $ij$ in algebraic form, where the literals $i$ and $j$ denote “the number of the row” and “the number of the column” respectively in a matrix.

$\left[\begin{array}{rr} 4 & -7 \\ -5 & 6 \\ \end{array}\right]$

Now, let’s learn how to represent the position of every entry in a matrix.

The entry $4$ belongs to the first row and also belongs to the first column. So, the position of an entry $4$ is the first row and first column, and it is written as $e_{11} \,=\, 4$ in mathematics.
The element $-7$ belongs to the first row and also belongs to the second column. So, the position of an element $-7$ is the first row and second column, and it is written as $e_{12} \,=\, -7$ in mathematics.
The entry $-5$ belongs to the second row and also belongs to the first column. So, the position of an element $-5$ is the second row and first column, and it is written as $e_{21} \,=\, -5$ in mathematics.
The element $6$ belongs to the second row and also belongs to the second column. So, the position of an entry $6$ is the second row and second column, and it is written as $e_{22} \,=\, 6$ in mathematics.