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Identity Matrix

The identity matrix is referred to as the multiplicative identity of matrices. Before going to learn what is an identity matrix, let us recall the meaning of identity in math. Identity is a mathematical quantity which when operated with some quantity leaves the same quantity. Let us consider the following examples.

The additive identity is 0 as adding any number to 0 gives the same number as the sum.
For example, 3 + 0 = 3, 0 + (-1) = -1, etc.
The multiplicative identity is 1 as multiplying any number with 1 gives the same number as the product.
For example, 3 × 1 = 3, 1 × (-1) = -1, etc.

In the same way, of course, we know that by adding the null matrix \(\left[\begin{array}{rr}0 & 0 \\ \\ 0 & 0 \end{array}\right]\) to any 2 × 2 matrix gives us the same matrix and hence a null matrix is called additive identity matrix. But what is the multiplicative identity with respect to matrices? The answer is the identity matrix and let us learn this in detail here.

1.	What is Identity Matrix?
2.	Identity Matrix of Different Orders
3.	Verification of Identity Matrix
4.	Properties of Identity Matrix
5.	Finding Inverse Matrix Using Identity Matrix
6.	Applications of Identity Matrix
7.	FAQs on Identity Matrix

What is Identity Matrix?

An identity matrix is a square matrix in which each of the elements of its principal diagonal is a 1 and each of the other elements is a 0. It is also known as the unit matrix. We represent an identity matrix of order n × n (or n) as I_n. Sometimes we denote this simply as I. The mathematical definition of an identity matrix is,

I_n (or) I = [a_ij]_{n × n}, where a_ij = 1 when i = j, and a_ij = 0 when i ≠ j.

An identity matrix in general is an identity with respect to multiplication. Thus, for any matrix A,

AI = IA = A

i.e., by multiplying any matrix A with the identity matrix of the same order, we get the same matrix as the product and hence the name "identity" for it.

Identity matrix definition

Identity Matrix of Different Orders

Here are some examples of identity matrices of different orders. Note that an identity matrix is a square matrix always.

2x2 identity matrix: I₂ = \(\left[\begin{array}{rr}1 & 0 \\ \\ 0 & 1 \end{array}\right]\).
3x3 identity matrix: I₃ = \(\left[\begin{array}{rr}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{array}\right]\).
4x4 identity matrix: I₄ = \(\left[\begin{array}{rr}1 & 0 & 0&0\\ 0 & 1 & 0&0\\ 0 & 0 & 1&0\\0&0&0&1\end{array}\right]\).

Verification of Identity Matrix

If I is an identity matrix and A is any matrix of the same order, then by the definition, AI = IA = A. Let us verify this by taking matrices of order 2 × 2.

A = \(\left[\begin{array}{rr}1 & 5 \\ \\-3 & 2 \end{array}\right]\) and I = \(\left[\begin{array}{rr}1 & 0 \\ \\ 0 & 1 \end{array}\right]\) (which is identity with respect to multiplication).

Verifying AI = A

AI = \(\left[\begin{array}{rr}1 & 5 \\ \\-3 & 2 \end{array}\right]\) \(\left[\begin{array}{rr}1 & 0 \\ \\ 0 & 1 \end{array}\right]\)

= \(\left[\begin{array}{rr}1(1)+5(0) & 1(0)+5(1) \\ \\ -3(1)+2(0) & -3(0)+2(1) \end{array}\right]\)

= \(\left[\begin{array}{rr}1 & 5 \\ -3 & 2 \end{array}\right]\)

= A

Since matrices don't need to be commutative with respect to multiplication, we have to verify IA = A as well.

Verifying IA = A

IA = \(\left[\begin{array}{rr}1 & 0 \\ \\ 0 & 1 \end{array}\right]\) \(\left[\begin{array}{rr}1 & 5 \\ -3 & 2 \end{array}\right]\)

= \(\left[\begin{array}{rr}1(1)+0(-3) & 1(5)+0(2) \\ \\ 0(1)+1(-3) & 0(5)+1(2) \end{array}\right]\)

= \(\left[\begin{array}{rr}1 & 5 \\ \\ -3 & 2 \end{array}\right]\)

= A

Thus, we have verified that AI = IA = A.

Similarly, you can try verifying the identity matrix of orders 3 × 3, 4 × 4, etc.

Properties of Identity Matrix

Here are the identity matrix properties based upon its definition.

The identity matrix is always a square matrix.
By multiplying an identity matrix with any other matrix results in the same matrix.
Every identity matrix is a diagonal matrix as only its principal diagonal's elements are nonzeros.
An identity matrix is symmetric as I^T= I.
Every identity matrix is a scalar matrix as all its principal diagonal's elements are equal and the rest of the elements are zeros.
The determinant of every identity matrix is 1.
The inverse of identity matrix is itself as I · I^-1= I^-1 · I = I.
Iⁿ = I, for any integer 'n'. i.e., the square of identity matrix is equal to itself, the cube of identity matrix is equal to itself, and so on.
By multiplying a matrix with its inverse, the result is an identity matrix
We can find the inverse of a matrix using the identity matrix (Let us see this in the next section).

Finding Inverse Matrix Using Identity Matrix

The inverse of a matrix A (which is written as A^-1) is a matrix B (and vice versa) if and only if AB = BA = I, where A, B, and I are the square matrices of the same order. Given A and B, it is easy to verify whether they are inverses of each other just by verifying whether AB = BA = I. But if a matrix A is given then how can we find its inverse B? We can find the inverse matrix of a matrix using the following steps:

Step 1: Write an augmented matrix with the given matrix adjoining it with the identity matrix of the same order and we separate these two matrices by a line.
Step 2: We apply row operations aiming to convert the left side matrix (which is A) to an identity matrix.
Step 3: The matrix that is left on the right side itself is our inverse matrix.

inverse matrix using identity matrix

We can see an example of finding the inverse matrix using these steps in the "Identity Matrix Examples" section below.

Applications of Identity Matrix

The identity matrix is used for various purposes in linear algebra. Here are the applications of the identity matrix.

An identity matrix is used to find the inverse of a matrix.
Also, an identity matrix is used to verify whether any two given matrices are inverses of each other.
An identity matrix is used to find the eigenvalues and eigenvectors.
An identity matrix is used while solving the system of equations using the elementary row operations.

Important Notes on Identity Matrix:

Here are some important points to note that are related to an identity matrix.

If you see an identity matrix without any specification of operation, then by default, it should be understood that it is an identity matrix with respect to multiplication.
To write an identity matrix of some order, first, write an empty matrix with the given order, write 1s in the place of elements of the principal diagonal, and finally write 0s in place of all other elements.
If AB = BA = I, then A and B are inverses of each other.
To find the inverse of a matrix, write it adjoining the identity matrix of the same order to it on its right side. Apply row operations to the entire augmented matrix aiming to make the left side matrix an identity matrix. Then the right side matrix will be the inverse of the given matrix.

☛ Related Topics:

Here are some topics that you may find interesting while reading about the identity matrix.

Identity Matrix Examples

Example 1: If I = \(\left[\begin{array}{rr}\cos x & 0 \\ \\ 0 & \sec x \end{array}\right]\) is an identity matrix of order 2 x 2, then find the value of x given that 0 ≤ x < π/2.

Solution:

We know that the I = \(\left[\begin{array}{rr}1 & 0 \\ \\ 0 & 1 \end{array}\right]\). Thus,

\(\left[\begin{array}{rr}\cos x & 0 \\ \\ 0 & \sec x \end{array}\right]\) = \(\left[\begin{array}{rr}1 & 0 \\ \\ 0 & 1 \end{array}\right]\)

Comparing the corresponding elements,

cos x = 1 and sec x = 1

These both can be true only when x = 0.

Answer: x = 0.
Example 2: For A = \(\left[\begin{array}{rr}1 & 2 & -1\\ 3 & 2 & 0\\ -4 & 0 & 2\end{array}\right]\) and I is the identity matrix of order 3 x 3, prove that A · I = I · A = A.

Solution:

Verifying A · I = A:

A · I = \(\left[\begin{array}{rr}1 & 2 & -1\\ 3 & 2 & 0\\ -4 & 0 & 2\end{array}\right]\) \(\left[\begin{array}{rr}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{array}\right]\)

= \(\left[\begin{array}{rr}1+0+0 & 0+2+0 & 0+0-1
\\ 3+0+0 & 0+2+0 & 0+0+0
\\ -4+0+0& 0+0+0& 0+0+2\end{array}\right]\)

= \(\left[\begin{array}{rr}1 & 2 & -1\\ 3 & 2 & 0\\ -4 & 0 & 2\end{array}\right]\)

= A

Verifying I · A = A:

I · A = \(\left[\begin{array}{rr}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{array}\right]\) \(\left[\begin{array}{rr}1 & 2 & -1\\ 3 & 2 & 0\\ -4 & 0 & 2\end{array}\right]\)

= \(\left[\begin{array}{rr}1+0+0 & 2+0+0 & -1+0+0\\
0+3+0 & 0+2+0 & 0+0+0\\
0+0-4 & 0+0+0 & 0+0+2\end{array}\right]\)

=\(\left[\begin{array}{rr}1 & 2 & -1\\ 3 & 2 & 0\\ -4 & 0 & 2\end{array}\right]\)

= A

Answer: We verified A · I = I · A = A.
Example 3: Find the inverse of \(\left[\begin{array}{rr}1 & 2 \\ \\ 3 & 4 \end{array}\right]\).

Solution:

Let A = \(\left[\begin{array}{rr}1 & 2 \\ \\ 3 & 4 \end{array}\right]\)

Step 1: Write an augmented matrix with A and I.

\(\left[\begin{array}{ll|ll}
1 & 2 & 1 & 0 \\ \\
3 & 4 & 0 & 1
\end{array}\right]\)

Step 2: Apply row operations to convert the left side matrix to an identity matrix.

Applying R₂ → R₂ - 3R₁,

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

Applying R₁ → R₁ + R₂,

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

Divide R₂ by -2,

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

Step 3: Here, the right side matrix is our inverse matrix. i.e.,

A^-1 = \(\left[\begin{array}{rr}-2 & 1 \\ \\ 3/2 & -1/2 \end{array}\right]\)

Answer: A^-1 = \(\left[\begin{array}{rr}-2 & 1 \\ \\ 3/2 & -1/2 \end{array}\right]\).

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Practice Questions on Identity Matrix

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FAQs on Identity Matrix

What is the Definition of an Identity Matrix in Linear Algebra?

An identity matrix, which is denoted by I, is a square matrix in which all elements of the principal diagonal are 1s and all the other elements are zeros. For any matrix A, AI = IA = A. It is also known as unit matrix.

How Do You Identify Identity Matrix?

If in a square matrix,

each element of the principal diagonal is a 1 and
all the other elements are zeros

then we say that it is an identity matrix.

What is Identity Matrix Squared?

The square of any identity matrix is itself. For example, \(\left[\begin{array}{rr}1 & 0 \\ \\ 0 & 1 \end{array}\right]\)\(\left[\begin{array}{rr}1 & 0 \\ \\ 0 & 1 \end{array}\right]\) = \(\left[\begin{array}{rr}1 & 0 \\ \\ 0 & 1 \end{array}\right]\).

What is the Formula of Identity Matrix?

An identity matrix is a matrix with 1s in its principal diagonal and 0s in all the other places. Thus, its formula is I_n (or) I = [a_ij]_{n × n}, where a_ij = 1 when i = j, and a_ij = 0 when i ≠ j.

What is the Identity Matrix of order 3?

In the identity matrix of order 3 × 3, all principal diagonal's elements are 1s and the other elements are 0s. It is given by \(\left[\begin{array}{rr}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{array}\right]\).

How To Find Inverse of a Matrix Using Identity Matrix?

To find the inverse of a matrix A, write A along with the identity matrix (I) of the same order in a matrix separating them by a line (such that A is on left and I is on right). Apply row operations to convert the left side matrix as I. Then the right side matrix is nothing but A^-1.

Why is It Called an Identity Matrix?

When an identity matrix (I) is multiplied with a matrix (A) of the same order, the product is the same matrix (A). i.e., AI = IA = A. So is the name (with respect to multiplication).

What is the Inverse of Identity Matrix?

The inverse of an identity matrix is itself. Because for any identity matrix I, we have I · I = I · I = I.

What is Identity Matrix Transpose?

The transpose of a matrix is obtained by writing its rows as columns (or columns as rows). The transpose of an identity matrix is itself. For example, If I = \(\left[\begin{array}{rr}1 & 0 \\ \\ 0 & 1 \end{array}\right]\) then its transpose is I^T = \(\left[\begin{array}{rr}1 & 0 \\ \\ 0 & 1 \end{array}\right]\) = I.

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