Math Doubts

Identity Matrix

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

Introduction

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.

identity 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.

  1. Oneness
  2. 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.

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

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

Examples

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.

Math Doubts

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

Maths Topics

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

Maths Problems

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

Learn solutions

Subscribe us

You can get the latest updates from us by following to our official page of Math Doubts in one of your favourite social media sites.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved