Learn More about this Identity Matrix Calculator

The identity matrix \(I\) is a very important matrix that has a very important property: If we multiply \(I\) by any matrix \(A\) (of suitable size), the matrix \(A\)gets unchanged by the multiplication.

In other words, the property the defines the identity matrix is

\[A I = I A = A\]

Now, we normally speak about "the" identity, when in fact there is an identity matrix for each integer \(n \ge 2\). So, given a size \(n\), we can construct the identity matrix for that specific size.

And that is what this what this calculator does: you provide a size \(n\) and the corresponding identity gets delivered to you.

Main Properties of the Identity Matrix

1. The identity matrix is a square matrix, in the sense that it has the same number of rows and columns
2. The identity matrix only has values different from zero at its diagonal
3. The diagonal contains only 1's
4. Multiplying the identity matrix I by another other matrix A (where the multiplication can be conducted) does not change its value. This is called the property of the identity matrix for the multiplication of matrices

How do you find an identity matrix?

This identity matrix calculator with steps can help you with that. So, what is the value of the identity matrix, or how do you calculate it? We first needs to specify the size \(n\) of the identity.

Step 1: Specify the desired size n of the identity matrix

Step 2: Then, the identity matrix is the matrix with \(n\) rows and \(n\) columns, which is defined as

\[ A_{i j} = \delta_{ij} \]

which means that \(A_{i j} = 1\) for when \( i = j\) and \(A_{i j} = 0\) for when \( i \ne j\).

Step 3: In layman terms, this is just a fancy way of saying that the identity matrix consists of 1's in the diagonal, and 0's outside of the diagonal.

Identity matrix Examples

The best way to understand about the identity matrix is to see some example, where you can understand how it works.

What is an identity matrix. Here is an example

For example, when \(n=2\), the identity matrix is that 2x2 matrix such that it has 1's in the diagonal and 0's outside of the diagonal. This looks like:

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

or when \(n=3\), the identity matrix is that 3x3 matrix such that it has 1's in the diagonal and 0's outside of the diagonal, which looks like:

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

Notation for the Identity

Some people will like to call \(I_2\) or \(I_{2x2}\) to the 2x2 identity. But you are fine calling it just \(I\), under the common understanding that there is an unambiguous size associated to that identity.

Interestingly enough, the identity matrix does not have any special property for the sum of matrices or for the subtraction of matrices, as it does for the multiplication.