Identity Matrix: Definition, Examples, and Properties

Identity Matrix: Definition, Examples, and Properties

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Vikram
Vikram Singh
Assistant Manager - Content
Updated on Apr 12, 2024 10:59 IST

Exploring the Identity Matrix: A Simple Guide – Dive into the basics of the identity matrix, a key element in linear algebra. We'll break down its definition, properties, and practical uses in a straightforward, easy-to-understand way. Perfect for students and enthusiasts looking to grasp this fundamental concept in mathematics.

identity matrix

A square matrix of order n x n with ones on the main diagonal and zeros elsewhere is known as an Identity Matrix. From solving a system of linear equations to representing identity transformation in 2D and 3D graphics applications, identity matrices play a significant role in linear algebra, calculus, and computer science.

Table of Content

What is the Identity Matrix?

Let A be any n x n matrix, then A is said to be an identity matrix if and only if 
Aij = 1, when i = j and 
Aij = 0, when i ≠ j
In simple terms, in an Identity Matrix, the entries on the principal diagonal (from the upper left to the bottom right) are equal to 1, and the remaining entries are equal to 0. 
It is represented by "I".

 

Examples of Identity Matrix

Identity Matrix of Order 1

identity matrix of order 1

Identity Matrix of Order 2

identity matrix of order 2

Identity Matrix of Order 3

identity matrix of order 3

Identity Matrix of Order 4

identity matrix of order 4

 

Properties of Identity Matrix

  • Square Matrix: An Identity matrix is always a square matrix with equal rows and columns.
  • Multiplicative Identity: Multiplying any square matrix with the identity matrix always returns the same matrix.
  • i.e., If A is any matrix and I is any square matrix, then A*I = A = I*A.
  • Determinant: The Determinant of the identity matrix is 1.
  • Inverse: The inverse of the identity matrix is an identity matrix.
  • EigenValues: All eigenvalues of the identity matrix are 1.
  • Symmetric Matrix: The identity matrix is symmetric, i.e., I = IT.
  • Orthogonal Matrix: The identity matrix is orthogonal, i.e., I*IT = I.
  • Trace: The trace of the identity matrix, which is equal to the sum of the diagonal elements, is equal to the order of the matrix.
  • Rank: The rank of the identity matrix is equal to the order of the matrix.
  • Power of Identity: Any power of the identity matrix is identity.

 

Operations on Identity Matrix

In this section, we will discuss two important operations on the Identity matrix, i.e., multiplication with the identity matrix and the inverse of the matrix.

Multiplication with the Identity Matrix

As we have already mentioned, the identity matrix serves as a multiplicative identity in matrix operations. So, let's take an example to understand better how this happens.

Let A and B be two matrices such that

A and B are two matrix of order 2 and 3

with the corresponding identity matrix

identity matrix of order 2 and 3

Now, multiplying A and I2

multiplying 2x2 matrix with identity matrix
multiplying 2x2 identity matrix with a 2x2 matrix

Hence, from above, we get A * I2 = A.

Now, let's see how to multiply a 3x3 matrix with the identity matrix.

multiplication of 3x3 matrix with the identity matrix
multiplication of 3x3 identity matrix

Hence, from above, we get B x I3 = B.

The inverse of Matrix using Identity Matrix

Here, we will see how to find the inverse of a 2x2 matrix using the identity matrix.

Let's take a 2x2 matrix.

let A be a 2x2 matrix

We will augment A with the identity matrix I to find the inverse of matrix A. Now, we will perform row operations to convert A into an identity matrix.

augmented matrix

Note: The operation will be applied to both sides of the augmented matrix.

Step-1: Make the element a11 = 1.
So, divide the first row by 4.

inverse matrix

Step-2: Make the element a21 = 0
So, multiply the first row by 3 and subtract it from the second row.

inverse matrix1

Step-3: Make the element a22 = 1
So, multiply the second row by 4.

inverse image 2

Step-4: Make the element a12 = 0
So, multiply the second row by 3/4 and subtract it from the first row.

inverse3

Hence, the inverse of A is

inverse of A

 

Real-Life Application of Identity Matrix

Domain Application Scenario-Based Example
Mathematics Solving Linear Equations Using the identity matrix to find the inverse of a matrix to solve for AX = B for .
Engineering Control Systems In state-space representation, using the identity matrix to model the state transition without external inputs.
Computer Science Algorithms and Data Structures Initializing transformation matrices in computer graphics for object manipulation in 3D space.
Physics Quantum Mechanics Representing the identity operator in finite-dimensional spaces to indicate no change in the quantum state.
Economics Markov Decision Processes Representing the initial state probabilities in consumer choice modelling before any transitions.
Robotics Movement and Positioning of Robot Arms Using the identity matrix as the base configuration for robot arm movement in kinematic equations.
Cryptography Encryption Algorithms Starting with the identity matrix for generating keys in matrix-based encryption techniques.

Conclusion

In this article, we have briefly discussed identity matrix, examples of an identity matrix, properties of an identity matrix, and operations on an identity matrix; and later, we have also discussed the real-life application of an identity matrix.  

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Vikram Singh
Assistant Manager - Content

Vikram has a Postgraduate degree in Applied Mathematics, with a keen interest in Data Science and Machine Learning. He has experience of 2+ years in content creation in Mathematics, Statistics, Data Science, and Mac... Read Full Bio