Inverse of Identity Matrix
2026-02-28 10:02 Diff
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Last updated on November 25, 2025

The identity matrix is unique because it acts like the number 1 in matrix multiplication. Its inverse is the identity matrix itself, since multiplying it by itself gives the same matrix. In this lesson, we’ll explore why this property holds true.

What is the Inverse of Identity Matrix?

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An identity matrix is a square matrix that has 1s on the main diagonal, i.e., from top-left to bottom-right, and 0s in all other places.

The inverse of the identity matrix is the identity matrix itself. This is because multiplying any identity matrix by itself results in the identity matrix, just like 1 × 1 = 1 in arithmetic.

Hence, the inverse of the identity matrix is the identity matrix itself. For example, an identity matrix of order 2 is:

I2 = 01 10

Inverse of Identity Matrix Of Order n

A formula for the inverse of any square matrix A is:

A-1 = 1A × adj(A)

For an identity matrix, In:

The determinant is always 1, |In| = 1.

Its adjoint is itself, adj(In) = In. 

In-1 = 11In = In

Therefore, the inverse of the identity matrix of order n is the same identity matrix.

Inverse of Identity Matrix Of Order 2

The identity matrix of order 2 is:

I2 = 01 10

Determinant: |I2| = 1

Adjoint: adj(I2) = 01 10

The inverse of the identity matrix of order 2:

I2-1 = 11I2 = I2

01 10 × 01 10 = 01 10

So, the inverse of a 2 × 2 identity matrix is the same 2 × 2 identity matrix.  

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Inverse of Identity Matrix Of Order 3

The matrix is I3 = 

 

1

0

0

   

0

1

0

   

0

0

1

 

Determinant: |I3| = 1

Adjoint: adj(I3) = I3

The inverse of the matrix is:

I3-1 = 11I3 = I3

Multiplying I3 by itself gives I3. Therefore, the inverse of the identity matrix of order 3 is equal to the identity matrix of order 3.

Common Mistakes and How to Avoid Them in Inverse of Identity Matrix

Students often make mistakes when dealing with the inverse of the identity matrix by confusing it with general matrix inversion. These are some of the common mistakes that we can avoid in the future. 

Real Life Applications of Inverse of Identity Matrix

The identity matrix is a special type of square matrix with 1s on the main diagonal and 0s in all other positions. One of the unique properties of the identity matrix is that its inverse is itself. Here are some of the real-life applications of the inverse of the identity matrix.

  • Computer Graphics: In 3D graphics, identity matrices are used as the starting points for transformations such as rotation, scaling, or translation. The inverse of the identity matrix is used to reset the transformations and return the objects to their original state. For example, when designing a video game, multiplying a 3D object by In-1 = In, leaves it unchanged. 
  • Robotics and Control Systems: Robots use identity matrices to represent the default position or no movement. The inverse of the identity matrix is used to reset the robot’s position during calculations of joint movements.
  • Programming and Algorithms: In programming tools like NumPy or MATLAB, identity matrices are often used as default inputs in matrix operations. Since the identity matrix doesn’t change other matrices when multiplied, it simplifies algorithm design and debugging.

Problem 1

Find the inverse of the identity matrix of order 3.

Okay, lets begin

I3-1 = 

 

1

0

0

   

0

1

0

   

0

0

1

 

Explanation

The identity matrix of order 3 is I3 =

 

1

0

0

   

0

1

0

   

0

0

1

 

The determinant is |I3| = 1.

The adjoint is adj(I3) = I3.

Using the formula:

I3-1 = 1I3 × adj(I3), we get

I3-1 = 11 × I3 = I3

Well explained 👍

Problem 2

Verify that the inverse of the identity matrix of order 2 is the same matrix.

Okay, lets begin

I2-1 = I2

Explanation

We have, I2 = 01 10

Multiply I2 by itself:

I2 × I2 = 01 10 × 01 10 = 01 10

Well explained 👍

Problem 3

Is the inverse of I2 = 01 10 is 10 01

Okay, lets begin

No, the inverse of I2 is the same I2 itself.

Explanation

The inverse of the identity matrix I2 is:

I2-1 = 01 10

The given matrix is incorrect because multiplying I2 by 10 01 does not return I2.

Well explained 👍

Problem 4

If the inverse of a matrix A is A-1, what is the inverse of the identity matrix In?

Okay, lets begin

 The inverse of In is In itself.

Explanation

The identity matrix acts like the number 1 in multiplication. Since In × In = In, it is its own inverse.

Well explained 👍

Problem 5

Find the inverse of the identity matrix of order 4

Okay, lets begin

The inverse of I4 is I4 

Explanation

The identity matrix of order n is:

The determinant of |I4| is 1

The adjoint of I4 is I4

So, by the inverse formula:

I4-1 = (1 / det) × adjoint = 11 × I4 = I4

Well explained 👍

FAQs on Inverse of Identity Matrix

1.What is the inverse of an identity matrix?

The inverse of an identity matrix is the same identity matrix itself. For example, I2 = 01 10 and I2-1 = 01 10.

2. Does every identity matrix have an inverse?

Yes, every identity matrix of any order has an inverse, and it is always the same identity matrix. 

3.What is the determinant of an identity matrix?

The determinant of any identity matrix In is always 1. 

4. Can a zero matrix have an inverse like the identity matrix?

No, the zero matrix does not have an inverse. Only the identity matrix and other invertible matrices have inverses.

5.Can non-square matrices have an identity matrix or an inverse?

No, identity matrices and their inverses only exist for square matrices. 

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.