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.

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 non-diagonal 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.

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.

Consider an identity matrix of order 2 given by, I₂ = \(= \left[\begin{array}{ccc} 1 & 0 \\
0 & 1
\end{array}\right] \). Now, the determinant of the identity matrix of order 2 is given by, |I₂| = 1 and adj(I₂) \(= \left[\begin{array}{ccc} 1 & 0 \\
0 & 1
\end{array}\right] \). Hence, the inverse of identity matrix of order 2 is,

I₂^-1 = (1/|I₂|) adj (I₂)

= (1/1) I₂

= I₂

Therefore, the inverse of identity matrix of order 2 is equal to the identity matrix of order 2.

Next, we will evaluate the inverse of identity matrix of order 3. Consider an identity matrix I₃ \(= \left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right] \)

Then determinant of the identity of order 3 is |I₃| = 1 and the adjoint of the matrix is adj (I₃) = I₃. Using the formula for the inverse of matrix, we have

I₃^-1 = (1/|I₃|) adj (I₃)

= (1/1) I₃

= I₃

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 I₂ is I₂ and I₃ is I₃

☛ Related Topics:

• Matrix formula
• How to Solve Matrices
• Transpose of 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 I₃ is I₃

What is the Multiplicative Inverse of Identity Matrix?

The multiplicative inverse of the identity matrix is the identity matrix itself.