# Singular (or degenerate) matrix

On this post you will find what a singular (or degenerate) matrix is and when a matrix is singular. We also show you several examples of singular matrices and, finally, we explain all the properties of this type of matrix.

## What is a singular matrix?

The definition of a singular matrix, also known as a degenerate matrix, is as follows:

A singular (or degenerate) matrix is a square matrix whose inverse matrix cannot be calculated. Therefore, the determinant of a singular matrix is equal to 0.

## When is a matrix singular?

You have to solve the determinant of the matrix to know when a matrix is singular:

• If the determinant of the matrix is equal to zero, the matrix is singular or non-invertible.
• If the determinant of the matrix is nonzero, the matrix is invertible.

In conclusion, calculating the determinant of a matrix is the easiest way to check the invertibility of a matrix.

However, if you want to find the value of the inverse of a matrix, you should see how to calculate the inverse of a matrix.

## Examples of singular matrices

Having seen the meaning of singular or degenerate matrix, let’s see some examples of singular matrices of different dimensions:

### Example of a 2×2 singular matrix

We can easily check that it is a singular matrix by calculating its determinant:

The solution of the determinant of the matrix of order 2 is equal to 0, so it is a singular matrix.

### Example of a 3×3 singular matrix

We must find the determinant of the matrix to prove that it is a non-invertible matrix:

The determinant of the matrix of order 3 results in 0, therefore, it is a singular matrix.

### Example of a 4×4 singular matrix

By calculating the determinant of the matrix it is proven that it is a singular matrix:

The determinant of the matrix of order 4 is null, so its inverse matrix does not exist.

If you have questions about the calculations of the determinants, you can consult how to calculate a determinant in our website.

## Properties of singular matrices

The characteristics of singular matrices are the following:

• At least two columns or two rows of a singular matrix are linear combination and are therefore linearly dependent.
• Any matrix that contains a row or column filled with zeros is a singular matrix.
• The rank of a singular or degenerate matrix is less than its size.
• The matrix product of a singular matrix multiplied by any other matrix results in another singular matrix. This condition can be deduced from the properties of the determinants:

• Similarly, the power of a singular matrix is equal to another singular matrix, regardless of the exponent to which it is raised.
• The transpose of a singular matrix gives another singular matrix, since the determinant of a transposed matrix is equivalent to the determinant of the untransposed matrix:

• Multiplying a singular matrix by a scalar does not change its condition as a degenerate matrix.
• The adjoint of a singular matrix is also singular.
• Triangular matrices and diagonal matrices are degenerate matrices if at least one element of their main diagonal is zero.
• Obviously, the null matrix is a singular matrix.
• In the same way, a nilpotent matrix is also a singular matrix.
• A system of linear equations associated with a singular matrix has no solution or has infinite solutions.
• Finally, a square matrix is singular if and only if it has at least one eigenvalue equal to 0.
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