Singular Matrix - Definition, Properties, Examples, Meaning - Cuemath

Singular Matrix

We determine whether a matrix is a singular matrix or a non-singular matrix depending on its determinant. The determinant of a matrix 'A' is denoted by 'det A' or '|A|'. If the determinant of a matrix is 0, then it is said to be a singular matrix. Why do we need to have a specific name for the matrices with determinant 0? Let us see.

Let us learn more about the singular matrix along with its definition, formula, properties, and examples.

1.	What is a Singular Matrix?
2.	Identifying a Singular Matrix
3.	Properties of Singular Matrix
4	Singular Matrix and Non-Singular Matrix
5.	Theorem to Generate Singular Matrices
6.	FAQs on Singular Matrix

What is a Singular Matrix?

A singular matrix is a square matrix if its determinant is 0. i.e., a square matrix A is singular if and only if det A = 0.We know that the inverse of a matrix A is found using the formula A-1 = (adj A) / (det A). Here det A (the determinant of A) is in the denominator. We are aware that a fraction is NOT defined if its denominator is 0. Hence A-1 is NOT defined when det A = 0. i.e., the inverse of a singular matrix is NOT defined. i.e., there does not exist any matrix B such that AB = BA = I (where I is the identity matrix).

From the above explanation, a square matrix 'A' is said to be singular if

• det A = 0 (which is also written as |A| = 0) (or)
• A-1 is NOT defined (i.e., A is non-invertible).

Identifying a Singular Matrix

To determine/identify whether a given matrix is singular we need to check for two conditions:

• check whether A is a square matrix.
• check whether det A = 0.

Here are few examples to find whether a given matrix is singular.

• A = \(\left[\begin{array}{rr}3 & 6 \\ \\ 2 & 4 \end{array}\right]\) is a singular matrix because it is a square matrix (of order 2 × 2) and det A (or) |A| = 3 × 4 - 6 × 2 = 12 - 12 = 0.
• A = \(\left[\begin{array}{rr}1 & 2 & 2 \\ 1 & 2 & 2\\ 3 & 2&-1 \end{array}\right]\) is a singular matrix because it is a square matrix (of order 3 × 3) and as det A (or) |A| = 0 (as the first two rows are equal).

Properties of Singular Matrix

Here are some singular matrix properties based upon its definition.

• Every singular matrix is a square matrix.
• The determinant of a singular matrix is 0.
• The inverse of a singular matrix is NOT defined and hence it is non-invertible.
• By properties of determinants, in a matrix, * if any two rows or any two columns are identical, then its determinant is 0 and hence it is a singular matrix. * if all the elements of a row or column are zeros, then its determinant is 0 and hence it is a singular matrix. * if one of the rows (columns) is a scalar multiple of the other row (column) then the determinant is 0 and hence it is a singular matrix.
• A null matrix of any order is a singular matrix.
• The rank of a singular matrix is definitely less than the order of the matrix. For example, the rank of a 3x3 matrix is less than 3.
• All rows and columns of a singular matrix are NOT linearly independent.

Singular Matrix and Non-Singular Matrix

A non-singular matrix, as its name suggests, is a matrix that is NOT singular. Thus, the determinant of a non-singular matrix is a nonzero number. i.e., a square matrix 'A' is said to be a non singular matrix if and only if det A ≠ 0. Then it is obvious that A-1 is defined. i.e., a non-singular matrix always has a multiplicative inverse. Thus, we can summarize the differences between the singular matrix and non-singular matrix as follows:

Singular Matrix	Non Singular Matrix
A matrix 'A' is singular if det (A) = 0.	A matrix 'A' is nonsingular if det (A) ≠ 0.
If 'A' is singular then A-1 is NOT defined.	If 'A' is nonsingular then A-1 is defined.
Rank of A < Order of A.	Rank of A = Order of A.
Some rows and columns are linearly dependent.	All rows and columns are linearly independent.
If 'A' is singular then the system of simultaneous equations AX = B has either no solution or has infinitely many solutions.	If 'A' is non singular then the system of simultaneous equations AX = B has a unique solution.
Example: \(\left[\begin{array}{rr}3 & 6 \\ \\ 1 & 2 \end{array}\right]\) is singular as \(\left|\begin{array}{rr}3 & 6 \\ \\ 1 & 2 \end{array}\right|\) = 3 × 2 - 1 × 6 = 6 - 6 = 0.	Example: \(\left[\begin{array}{rr}3 & 2 \\ \\ 1 & -2 \end{array}\right]\) is non-singular as \(\left|\begin{array}{rr}3 & 2\\ \\ 1 & -2 \end{array}\right|\) = 3 × -2 - 1 × 2 = -6 - 2 = -8 ≠ 0.

Theorem to Generate Singular Matrices

There is one important theorem on singular matrix that can actually be used to generate a singular matrix and the theorem says: "The product of two matrices A = [A]n × k and B = [B]k × n (where n > k) is a matrix AB of order n × n and is always singular". By this theorem:

• The product AB of two matrices A of order n × 1 and B of order 1 × n is singular always.
• The product AB of two matrices A of order n × 2 and B of order 2 × n is also singular, etc.

Using this theorem, one can generate a singular matrix by multiplying two randomly generated matrices of orders n × k and k × n where n > k.

☛ Related Topics:

• Diagonal Matrix
• Orthogonal Matrix
• Inverse of 3x3 Matrix
• Symmetric Matrix