The identity matrix and its properties

The identity matrix is a square matrix that has 1’s along the main diagonal and 0’s for all other entries. This matrix is often written simply as \(I\), and is special in that it acts like 1 in matrix multiplication. In this lesson, we will look at this property and some other important idea associated with identity matrices.

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The identity matrix is always a square matrix

While we say “the identity matrix”, we are often talking about “an” identity matrix. For any whole number \(n\), there is a corresponding \(n \times n\) identity matrix. These matrices are said to be square since there is always the same number of rows and columns.

a 2 x 2 and a 3 x3 identity matrix

To prevent confusion, a subscript is often used. So in the figure above, the \(2 \times 2\) identity could be referred to as \(I_2\) and the \(3 \times 3\) identity could be referred to as \(I_3\).

Multiplying any matrix by the identity results in the matrix itself

When working with matrix multiplication, the size of a matrix is important as the multiplication is not always defined. Therefore for an \(m \times n\) matrix \(A\), we say:

multiplication property with the identity matrix

This shows that as long as the size of the matrix is considered, multiplying by the identity is like multiplying by 1 with numbers. For example, consider the following matrix.

example of a 2 by 4 matrix

This is a \(2 \times 4\) matrix since there are 2 rows and 4 columns. You can verify that \(I_2 A = A\):

left hand multiplication by the identity

and \(A I_4 = A\):

right hand multiplication by the identity

With other square matrices, this is much simpler. Consider the example below where \(B\) is a \(2 \times 2\) matrix. Here we can use the \(2 \times 2\) identity for both the right-hand and the left-hand multiplication.

multiplying a square matrix by the identity

The product of two inverse matrices is always the identity

One concept studied heavily in mathematics is the concept of invertible matrices, which are those matrices that have an inverse. By definition, when you multiply two matrices that are inverses of each other, then you will get the identity matrix. Consider the following matrices:

two inverse matrices

For these matrices, \(AB = BA = I\), where \(I\) is the \(2 \times 2\) identity matrix.

product of two inverse matrices is the identity matrix, as shown in this example

Therefore \(A\) and \(B\) are inverse matrices. You can study this idea more here: inverse matrices.

Summary

The identity matrix is a fundamental idea when working with matrices – whether you are working with just multiplication, inverses, or even solving matrix equations. As you study these types of topics, be sure that you have a fundamental understanding of this matrix.

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