Identity Matrix

What is Identity Matrix?

The identity matrix is known as the matrix that is in the form of the n × n square matrix in which the diagonal contains the ones and all the other elements are zeros. It is also referred to as a unit matrix or an elementary matrix. It is denoted as In or just I, wherein n is the size of the square matrix. To explain the identity matrix definition part by part, let us start by reminding you that the square matrix refers is the matrix that contains the same amount of the rows and the columns. The order of the matrix comes from its dimensions, and the main diagonal is the array of the elements that are inside the matrix that form the inclined line starting from the top left corner and extending to the bottom right corner. Given the characteristics of the identity matrix, you can also conclude that these type of matrices are also called as diagonal matrices. In this article, we will learn about what is an identity matrix, the determinant of identity matrix, identity matrix properties, the identity matrix in c, and learn about the identity matrix example.

Identity Matrix Definition

An identity matrix refers to a type of the square matrix in which its diagonal entries are equal to 1 and the off-diagonal entries are equal to 0.


Identity matrices play a vital role in the linear algebra. In particular, their role in the matrix multiplication is similar to the role that is played by the number 1 when it comes to the multiplication of the real numbers:


  1. The real number remains unchanged if it is multiplied by 1

  2. The matrix remains unchanged if it is multiplied by an identity matrix

Identity Matrix Properties

Let us discuss the properties of the identity matrix.

  1. Identity matrix is always in the form of a square matrix.

The identity matrix is called a square matrix because it has the same number of the rows and the columns. For any given whole number n, the identity matrix is given by n x n.

  1. Multiplying a given matrix with the identity matrix would result in the matrix itself.

Since the multiplication is not always defined, the size of the matrix matters when you work on the matrix multiplication.


For example, for the given m x n matrix C, you get


C = \[\begin{bmatrix} 1  & 2 & 3 & 4\\ 5 & 6 & 7  & 8 \end {bmatrix}\]


The above matrix is a 2 x 4 matrix since it contains 2 rows and 4 columns.


  1. When multiplying two inverse matrices, you would get an identity matrix.


If you multiply two matrices that are inverses of each other you would get an identity matrix.

Identity Matrix Example

Some examples of identity matrices are as follows:

The 2 x 2 identity matrix is given by


I = \[\begin{bmatrix} 1  & 0 \\ 0 & 1 \end {bmatrix}\]


The identity matrix of order 3 is represented in the following manner:


I = \[\begin{bmatrix} 1  & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1\end {bmatrix}\]

Solved Examples


Example 1

Write an identity matrix of the order 4

Solution:

The identity matrix of the order 4 x 4 is given as

I = \[\begin{bmatrix} 1  & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0  & 1 & 0\\ 0 & 0 & 0 & 1 \end {bmatrix}\]


Example 2

Determine if the given matrix is an identity matrix or not.

C = \[\begin{bmatrix} 1  & 0  & 0 \\ 0  & 0 & 1 \end {bmatrix}\]

Solution:

No, the given matrix is not an identity matrix since it is not a square matrix. The number of rows is not equal to the number of columns. The given matrix is of the order 2 x 3.

FAQ (Frequently Asked Questions)

1. What do you mean by an identity matrix?

Matrix multiplication is a type of a binary operation. Take two elements from a given set and then follow some rules and combine them together to some other element of the given set. Other examples of the binary operations include the addition of the real numbers and the multiplication of the real numbers.


Now, for some of the binary operations, we have what is called an identity element. This is a special element that leaves things alone under that particular operation. In addition, the identity is 0. Any number when added to 0 results in the same number. For the multiplication of numbers, the identity is 1. Any number multiplied by 1 results in the same number.


The matrix multiplication also contains an identity element. It is the matrix that leaves another matrix alone when it is multiplied by it. It acts just like the multiplication of the real numbers by 1.

2. How to find the determinant of a rectangular matrix such as a 2 x 3 matrix?

You cannot find the determinant of a rectangular matrix because determinant is only defined for the square matrices.


The determinant is said to represent the size of the n-dimensional hyperspace that is occupied by the n-dimensional rectangular parallelepiped which is having the column vectors of the square matrix in the form of its sides. 


Thus, in the case of a wide rectangular matrix such as 2 x 3, it does not make any sense to define the determinant, since there are three 2-dimensional vectors that do not form a parallelogram when it is drawn from the origin in the cartesian coordinate system.


Also, in the case of the tall rectangular matrix such as 3 x 2 you have two 3-dimensional vectors that cannot form the sides of a parallelepiped.