Singular Matrix

A singular matrix is a square matrix (i.e., a matrix where the number of rows is equal to the number of columns) whose determinant is zero.

• This means it can't be inverted, meaning you can't multiply it by another matrix to get the identity matrix.
• There is no matrix B such that when you multiply matrix A by B, you get the identity matrix I (i.e., AB = I).

Why is the Inverse of a Singular Matrix Not Defined

We know that the formula to determine the inverse of a matrix is equal to the adjoint of the matrix divided by the determinant, i.e.,

A^{-1} = \frac{\operatorname{adj}(A)}{|A|}

From the definition of a singular matrix, we know that ∣A∣ = 0. Since the determinant of A is zero, the matrix is singular, meaning its inverse does not exist.

Various examples of singular matrices are:

• \left[\begin{array}{ccc} 1 & 2 \\ 4 & 8\end{array}\right],

• \left[\begin{array}{ccc} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{array}\right],

• \left[\begin{array}{ccc} 1 & 0 & 0\\ 0 & 0 & 0\\ 7 & 8 & 9 \end{array}\right],

Theorem

Let A be a matrix of order n × k and B be a matrix of order k × n, where n>k. Then, the product AB, which is an n × n matrix, is always singular.

This means that if you multiply any n × k matrix by any k × n matrix, with n > k, the resulting square matrix will always be singular (i.e., its determinant will be zero). The intuition is that the rank of the product AB cannot exceed k, which is less than n, so AB cannot be full-rank and thus must be singular.

To generate a singular matrix of size n × n:

• Randomly generate a matrix A of size n × k (with k < n).
• Randomly generate a matrix B of size k × n.
• Compute AB. The resulting matrix is guaranteed to be singular

Example: If n = 3, k = 1.

Solution:

Let A = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}_{(3\times1),} \text{and,} \ B = \begin{bmatrix} 4 & 5 & 6 \end{bmatrix}_{ (1 \times 3)}

Then the product (AB) is :

AB = A \cdot B = \begin{bmatrix}1 \\2 \\3\end{bmatrix}\begin{bmatrix}4 & 5 & 6\end{bmatrix}= \begin{bmatrix}1 \cdot 4 & 1 \cdot 5 & 1 \cdot 6 \\2 \cdot 4 & 2 \cdot 5 & 2 \cdot 6 \\3 \cdot 4 & 3 \cdot 5 & 3 \cdot 6\end{bmatrix}= \begin{bmatrix}4 & 5 & 6 \\8 & 10 & 12 \\12 & 15 & 18\end{bmatrix}

AB is 3×3, but the rows are linearly dependent → Singular

Properties

The following are the properties of the Singular Matrix:

• Every singular matrix must be a square matrix, i.e., a matrix that has an equal number of rows and columns.
• The determinant of a singular matrix is equal to zero.
• As the determinant of a singular matrix is zero, its inverse is not defined.
• A zero matrix of any order is a singular matrix, as its determinant is zero.
• In a singular matrix, some rows and columns are linearly dependent.
• The rank of a singular matrix will be less than the order of the matrix, i.e., Rank (A) < Order of A.
• A matrix that has any two rows or any two columns identical is singular, as the determinant of such a matrix is zero.
• When a row or column's elements in a matrix are all zeros, then the matrix is singular, as its determinant is zero.
• When one row (or column) of a matrix is a scalar multiple of another row (or column), then the matrix is singular. It's with Matrices. Its determinant is zero. 

Singular vs Non-Singular Matrix

Let us consider that A and B are two square matrices of order "n × n".

• If AB = BA = I, where I is an identity or unit matrix of order n, then B is said to be the inverse matrix of A.
• Matrix A is non-singular (i.e., it has an inverse) if it satisfies this condition.
• If A does not have an inverse (i.e., no matrix B exists such that AB = BA = I), then A is called a singular matrix.

Thus:

• Non-singular matrix: Has an inverse.
• Singular matrix: Does not have an inverse.

Differences between the Singular Matrix and Non-Singular Matrix can be understood using the table given below.

Identifying a Singular Matrix

Follow the conditions given below to determine whether the given matrix is singular or not.

• Determine whether the given matrix is square or not.
• If the given matrix is square, then find the determinant of the matrix.

⇒ If |A|= 0, then the given matrix is singular.

⇒ If |A|≠0, then the given matrix is non-singular.

➢Practice: Solved Examples