Inverse of Identity Matrix
The inverse of identity matrix of order n is the identity matrix itself. According to the definition of inverse of a matrix, the product of a matrix and its inverse is equal to the identity matrix of the same order. Since the product of the identity matrix with itself is equal to the identity matrix, therefore the inverse of identity matrix is the identity matrix itself.
In this article, we will determine the inverse of the identity matrix of orders 2, 3 and n using the formula, and solve a few examples based on it for a better understanding of the concept.
1.  What is Inverse of Identity Matrix? 
2.  Inverse of Identity Matrix Of Order n 
3.  Inverse of Identity Matrix Of Order 2 
4.  Inverse of Identity Matrix Of Order 3 
5.  FAQs on Inverse of Identity Matrix 
What is Inverse of Identity Matrix?
The inverse of an identity matrix is the identity matrix itself of the same order, that is, the same number of rows and columns. An identity matrix is a square matrix with all main diagonal elements equal to 1 and nondiagonal elements are equal to 0. To determine the inverse of identity matrix, we multiply it with a matrix such that the product is equal to the identity matrix. So, the matrix whose product with the identity matrix gives an identity matrix is the identity matrix itself. Hence, the inverse of identity matrix is the identity matrix itself.
Inverse of Identity Matrix Of Order n
We know that the formula to determine the inverse of a matrix A is A^{1} = (1/A)adj A. Let us consider an identity matrix I_{n} of order n. Now, the determinant of an identity matrix is always equal to1 and its adjoint is given by, adj I_{n} = I_{n}. Next, using the formula, the inverse of identity matrix of order n is given by,
I_{n}^{1} = (1/I_{n}) adj (I_{n})
= (1/1) I_{n}
= I_{n}
Therefore, the inverse of identity matrix of order n is equal to the identity matrix of order n.
Inverse of Identity Matrix Of Order 2
Consider an identity matrix of order 2 given by, I_{2} = \(= \left[\begin{array}{ccc} 1 & 0 \\
0 & 1
\end{array}\right] \). Now, the determinant of the identity matrix of order 2 is given by, I_{2} = 1 and adj(I_{2}) \(= \left[\begin{array}{ccc} 1 & 0 \\
0 & 1
\end{array}\right] \). Hence, the inverse of identity matrix of order 2 is,
I_{2}^{1} = (1/I_{2}) adj (I_{2})
= (1/1) I_{2}
= I_{2}
Therefore, the inverse of identity matrix of order 2 is equal to the identity matrix of order 2.
Inverse of Identity Matrix Of Order 3
Next, we will evaluate the inverse of identity matrix of order 3. Consider an identity matrix I_{3} \(= \left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right] \)
Then determinant of the identity of order 3 is I_{3} = 1 and the adjoint of the matrix is adj (I_{3}) = I_{3}. Using the formula for the inverse of matrix, we have
I_{3}^{1} = (1/I_{3}) adj (I_{3})
= (1/1) I_{3}
= I_{3}
Therefore, the inverse of identity matrix of order 3 is equal to the identity matrix of order 3.
Important Notes on Inverse of Identity Matrix
 The inverse of identity matrix is the identity matrix itself of the same order.
 The inverse of identity matrix is a diagonal matrix and symmetric matrix.
 The inverse of identity matrix I_{2} is I_{2} and I_{3} is I_{3}
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Inverse of Identity Matrix Examples

Example 1: Determine the inverse of a scalar matrix kI_{2} using the inverse of identity matrix.
Solution: Scalar matrix kI_{2} \(= \left[\begin{array}{ccc} k & 0 \\
0 & k
\end{array}\right] \)We can write this matrix as kI_{2} \(= k\left[\begin{array}{ccc} 1 & 0 \\
0 & 1
\end{array}\right] \)As the inverse of identity matrix is the identity matrix itself, therefore the inverse of kI_{2} is kI_{2}.
Answer: Inverse of a scalar matrix kI_{2} is kI_{2}

Example 2: What is the inverse of identity matrix of order 12.
Solution: The order of the identity matrix does not change the formula for the inverse of the identity matrix.
Therefore, the inverse of identity matrix of order 12 is the identity matrix of order 12.
Answer: Inverse of identity matrix of order 12 is I_{12}.
FAQs on Inverse of Identity Matrix
What is Inverse of Identity Matrix in Matrix Algebra?
Since the product of the identity matrix with itself is equal to the identity matrix, therefore the inverse of identity matrix is the identity matrix itself.
How to Find the Inverse of Identity Matrix?
To find the inverse of identity matrix, we can use the formula for the inverse of a matrix A is A^{1} = (1/A)adj A, where A can be substituted with the identity matrix.
What is the Inverse of Identity Matrix of Order n?
The inverse of identity matrix of order n I_{n} is given by I_{n} itself.
What is the Inverse of Identity Matrix of Order 3?
The inverse of identity matrix of 3 × 3 I_{3} is I_{3}
What is the Multiplicative Inverse of Identity Matrix?
The multiplicative inverse of the identity matrix is the identity matrix itself.
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