Identity Matrix

An identity matrix is a type of square matrix with 1s on the main diagonal and 0s in all other positions. We call it an identity matrix because when we multiply it by other matrices of the same size, it won't change that matrix.


It works like how multiplying 1 by any number does not change the number. If A is a matrix, then: A × I = I × A = A. The identity matrix is also known as the multiplicative identity for matrices. 


2 × 2 identity matrix: 

 

\(I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\)

A 3 × 3 identity matrix is written as:


\(I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\)
 

The properties of the identity matrix are important when solving equations, transforming shapes, or working with matrix algebra. Given below are some of the properties of the identity matrix.

 

• Square Matrix: An Identity matrix is always a square matrix because it has an equal number of rows and columns. 2 × 2, 3 × 3, etc., are the orders of an identity matrix.
  
   
• Multiplicative Identity: Multiplying any matrix with an identity matrix results in the same matrix. If A is a matrix and I is the identity matrix, A × I = I × A = A.
  
   
• Determinant is 1: The determinant is a value associated with square matrices. For an identity matrix, the determinant is always equal to 1, regardless of its size.
  
   
• Inverse itself: The inverse of the matrix is the same identity matrix. I -1 = I.
  
   
• Eigenvalues are 1: When we find the eigenvalues for an identity matrix, all of them are 1.
  
   
• Symmetric Matrix: A matrix is called symmetric if it remains the same when reflected across its diagonal. The identity matrix is always symmetric because its transpose equals itself, that is, I = I^T.
  
   
• Orthogonal Matrix: The identity matrix is an orthogonal matrix, which means that when we multiply it by its transpose, the result is still the identity matrix. I  = I^T I = I.
  
   
• Trace Equals Matrix Order: The trace of a matrix is the sum of its main diagonal elements. In an identity matrix, all diagonal elements are 1, so the trace equals the number of rows (or columns) of the matrix.
  
   
• Rank Equals Order: The rank of a matrix refers to the number of linearly independent rows or columns. In an identity matrix, every row is unique and independent, so the rank is equal to its order. 
  
   
• Any Power is Identity: If we multiply an identity matrix many times by itself, it always stays the same. So, I² = I, I³ = I, and so on.
   

The order of an identity matrix refers to the dimensions of the matrix. Since an identity matrix is a square matrix, it is always in the form of n × n. The identity matrix is always square. Here are some of the orders of an identity matrix. 


2 × 2 identity matrix: 


\(I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\)

3 × 3 identity matrix:

\(I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\)

 

4 × 4 identity matrix:

 

\(I_4 = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}\)

 

5 × 5 identity matrix:


\(I_5 = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{bmatrix}\)

The inverse of an identity matrix works similarly to the inverse of numbers. Multiplying a number and its reciprocal gives 1; multiplying a matrix by its inverse is called the identity matrix.


If A is a matrix, its inverse is denoted by A-1 and multiplying them gives the identity matrix: A × A-1 = I.


Here, I is the identity matrix. The identity matrix looks like: 


\(I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\). Given below are the steps to find the inverse of a matrix using the identity matrix.

Step 1: Write the given matrix next to the identity matrix, and draw a line to show that they are side by side.


Example: \(A = \begin{bmatrix} 5 & 2 \\ 3 & 1 \end{bmatrix}\).


Draw the matrix A left side and add the identity matrix of the same order on the right side.

 

\(\left[ \begin{array}{cc|cc} 2 & 1 & 1 & 0 \\ 5 & 3 & 0 & 1 \end{array} \right]\)

Step 2: Apply row operations to change the left side of an augmented matrix into an identity matrix.
To make the first element as 1 in row 1, we must divide row 1 (R1) by 2.
R1 = R1 ÷ 2 
 

\(\left[ \begin{array}{cc|cc} 1 & 2.5 & 0.5 & 0 \\ 5 & 3 & 0 & 1 \end{array} \right]\)

 

We want to make the first element 0 in row 2.
R2 = R2  - R1

 

\(\left[ \begin{array}{cc|cc} 1 & 2.5 & 0.5 & 0 \\ 0 & 0.5 & -0.5 & 1 \end{array} \right]\)

 

Make the second element in row 2 as 1. Divide row 2 by 0.5.
R2 = R2 ÷ 0.5

 

\(\left[ \begin{array}{cc|cc} 1 & 2.5 & 0.5 & 0 \\ 0 & 1 & -1 & 2 \end{array} \right]\)

 

Make the second element from row 1 as 0. To make it, multiply 2.5 by row 2 and subtract row 1.
R1 = R1 - 2.5 × R2

 

\(\left[ \begin{array}{cc|cc} 1 & 0 & 3 & -5 \\ 0 & 1 & -1 & 2 \end{array} \right]\)

Step 3: The left side is the identity matrix, and the right side is the inverse of A. Therefore, the inverse of A is:


\(A^{-1} = \begin{bmatrix} -1 & 3 \\ 2 & -5 \end{bmatrix}\)
 

An identity matrix, denoted by I, is a square matrix with 1s on the diagonal and 0s elsewhere. An n × n identity matrix is I_n. Two matrices whose product equals I are called inverses.


The format of I_{n × n} matrix is:


\(\begin{bmatrix} 1 & \cdot & \cdot & \cdot & \cdot \\ 0 & 1 & \cdot & \cdot & \cdot \\ \cdot & \cdot & 1 & \cdot & \cdot \\ \cdot & \cdot & \cdot & 1 & \cdot \\ \cdot & \cdot & \cdot & \cdot & 1 \end{bmatrix}\)

The two main operations used in the identity matrix are:

 

• Multiplying with the Identity Matrix
  
   
• Inverse of a Matrix using Identity Matrix
  
   

Multiplying with the Identity Matrix:


Multiplying any square matrix by the identity matrix gives the same matrix. Let’s understand how to multiply matrices with an identity matrix through an example. 


Example:


\(A = \begin{bmatrix} 4 & 3 \\ 3 & 2 \end{bmatrix}\)


We need to multiply the given matrix by an identity matrix of the same order. The identity matrix is:
I \(= \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\)


Multiply the matrices,


\(A \times I = \begin{bmatrix} 4 & 3 \\ 3 & 2 \end{bmatrix} \times \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 4 & 3 \\ 3 & 2 \end{bmatrix} = A\)


Therefore, multiplying any matrix by an identity matrix results in the same matrix. 


Inverse of a Matrix using Identity Matrix:


To find the inverse of a matrix, we use a method that involves row operations and the identity matrix, follow the steps given below:

 

Step 1: Create an augmented matrix by placing the identity matrix of the same size next to the given matrix.


Step 2: Use row operators to change the left side of an augmented matrix into an identity matrix B


Step 3: Once the left side becomes the identity matrix, the right side will be the inverse of the original matrix. 
 

The identity matrix is a special square matrix that acts like the number 1 in matrix operations. It is very important in algebra and linear equations because it leaves other matrices unchanged when multiplied. Knowing its key properties and tricks can make calculations faster and help you solve problems more easily.

 

• A square matrix where all diagonal elements are 1 and all off-diagonal elements are 0.
  
   
• Whenever you see A × I or I × A, you can directly write A without doing full multiplication.
  
   
• Multiplying by the identity matrix does not change the original matrix.
  
   
• Transposing an identity matrix gives the same matrix because it is symmetric along the diagonal.
  
   
• Multiplying by a number only scales the diagonal elements.

The identity matrix has many practical uses in real-world situations, especially in fields like computer graphics, engineering, and data science. Listed below are some of the real-life applications of the identity matrix.

 

• Engineering: In control systems, the identity matrix represents a system that remains unchanged when no input is applied. Used in signal processing to maintain the state of a system, ensuring outputs remain consistent in the absence of external forces.
  
   
• Computer Animation and Graphics: In 3D graphics, identity matrices are used as starting points for transformations like translation, rotation, and scaling. They ensure that objects maintain their original position and orientation before applying transformations.
  
   
• Cryptography: In matrix based encryption methods, the identity matrix is used as a reference for constructing key matrices. It ensures that only the correct inverse matrix can decode the encrypted message, maintaining the security of the information.
  
   
• Robotics: Identity matrices are used in robot kinematics to represent a robot’s initial position or neutral configuration. They are essential in calculating movement and transformations of robotic arms without altering the starting state.
  
   
• Architecture: In structural modeling, identity matrices help in simulating forces and maintaining reference positions of elements. They act as baseline matrices when performing transformations or stress analysis on building components.