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101 LearnersLast updated on November 25, 2025
Inverse of Identity Matrix
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?
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.
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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.
Inverse of Identity Matrix Of Order 3
The matrix is I3 =
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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.
Mistake 1
Thinking that the inverse of the identity matrix is different
Remember that the identity matrix has a special property: multiplying it by any compatible matrix leaves the other matrix unchanged. Since the identity matrix already behaves like the "1" of matrix multiplication, its inverse is itself. So, the inverse of an identity matrix is the same identity matrix, not a different one.
Mistake 2
Confusing identity and zero matrix
Some students get confused with the identity matrix and the zero matrix. A zero matrix does not have an inverse like the identity matrix. Check the diagonal, the identity matrix always has 1s on its diagonal, not 0s. The identity matrix looks like, 01 10 and the zero matrix looks like: 00 00.
Mistake 3
Mistakes in writing the identity matrix
Sometimes students incorrectly write an identity matrix with values other than 1 on the diagonal, or place the 0s in the wrong positions off the diagonal. Write the 1s only on the main diagonal from top-left to bottom-right, and all the other elements are 0s. Writing I2 = 01 11 is wrong.
Mistake 4
Confusing inverse and transpose
The transpose of a matrix is found by flipping its rows and columns, while the inverse is a matrix that, when multiplied by the original, gives the identity matrix. These are not the same operation. To avoid confusion, remember that transpose changes the position of elements and inverse undoes the matrix through multiplication.
Mistake 5
Misunderstanding the role of determinant
Students think that for an identity matrix, they have to find the determinant every time. But for an identity matrix, the determinant is always 1. Memorize that |In| = 1.
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.
Solved Examples of Inverse of Identity Matrix
Problem 1
Find the inverse of the identity matrix of order 3.
I3-1 =
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Explanation
The identity matrix of order 3 is I3 =
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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
Problem 2
Verify that the inverse of the identity matrix of order 2 is the same matrix.
I2-1 = I2
Explanation
We have, I2 = 01 10
Multiply I2 by itself:
I2 × I2 = 01 10 × 01 10 = 01 10
Problem 3
Is the inverse of I2 = 01 10 is 10 01
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.
Problem 4
If the inverse of a matrix A is A-1, what is the inverse of the identity matrix In?
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.
Problem 5
Find the inverse of the identity matrix of order 4
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
FAQs on Inverse of Identity Matrix
1.What is the inverse of an identity matrix?
2. Does every identity matrix have an inverse?
3.What is the determinant of an identity matrix?
4. Can a zero matrix have an inverse like the identity matrix?
5.Can non-square matrices have an identity matrix or an inverse?
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.
