A matrix that consists of ones as diagonal elements and zeros as non-diagonal elements is called an identity matrix.

In a special case, each element in the main diagonal (or leading diagonal) can be one and the remaining non-diagonal entries can be zeros in a matrix.

$I$ $\,=\,$ $\begin{bmatrix} 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \color{red} \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \\ \end{bmatrix}$

There are two popular meanings for the word “identity” in English language.

- Oneness
- Unity

Hence, this type of matrix is called an identity matrix, and simply denoted by $I$ in mathematics. It is also called a unit matrix.

An identity matrix is basically a square matrix.

- Diagonal elements are $1$s.
- Non-diagonal elements are $0$s.

Due to these two reasons, a unit matrix is a diagonal matrix principally and also a scalar matrix.

The following three examples help you to understand how to express the identity matrices of different orders.

$I_2$ $\,=\,$ $\begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} $

It is a matrix of the order $2 \times 2$ but it is an identity matrix and also a square matrix. Hence, it is known as an identity matrix of order $2$. It is simply denoted by $I_2$ in matrix.

$I_3$ $\,=\,$ $\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end{bmatrix} $

It is a matrix of the order $3 \times 3$. It is an identity matrix and also a square matrix. Therefore, it is called an identity matrix of order $3$ and simply denoted by $I_3$ in mathematics.

$I_4$ $\,=\,$ $\begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ \end{bmatrix} $

It is a matrix of the order $4 \times 4$. It is an identity matrix and also a square matrix. So, it is called an identity matrix of order $4$ and simply denoted by $I_4$ in mathematics.

In this way, an identity matrix of any order can be expressed in mathematics.

$I_n$ $\,=\,$ $\begin{bmatrix} 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \color{red} \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \\ \end{bmatrix}$

If the number of entries in each row is $n$ and the number of elements in each column is $n$, then it is called a matrix of the order $n \times n$. It is an identity matrix and also a square matrix. Hence, it is called an identity matrix of order $n$ and simply denoted by $I_n$ in mathematics.

Latest Math Topics

Jan 06, 2023

Jan 03, 2023

Jan 01, 2023

Dec 26, 2022

Dec 08, 2022

Latest Math Problems

Jan 31, 2023

Nov 25, 2022

Nov 02, 2022

Oct 26, 2022

A best free mathematics education website for students, teachers and researchers.

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Learn how to solve the maths problems in different methods with understandable steps.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved