Identity Matrix: Definition, Examples, and Properties

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.

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?
• Examples of Identity Matrix
• Properties of Identity Matrix
• Operations on Identity Matrix
  • Multiplication of Identity Matrix
  • Inverse of Matrix using Identity Matrix
• Real-Life Application of Identity Matrix

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 
A_ij = 1, when i = j and 
A_ij = 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 2

Identity Matrix of Order 3

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 = I^T.
• Orthogonal Matrix: The identity matrix is orthogonal, i.e., I*I^T = 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

with the corresponding identity matrix

Now, multiplying A and I₂

Hence, from above, we get A * I₂ = A.

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

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.

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.

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

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

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

Hence, the inverse of A is

Real-Life Application of Identity Matrix

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.  

Keep Learning!!