Singular Matrix
We determine whether a matrix is a singular matrix or a nonsingular 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.
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 noninvertible).
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 noninvertible.
 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 NonSingular Matrix
A nonsingular matrix, as its name suggests, is a matrix that is NOT singular. Thus, the determinant of a nonsingular 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 nonsingular matrix always has a multiplicative inverse. Thus, we can summarize the differences between the singular matrix and nonsingular 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 nonsingular 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:
Singular Matrix Examples

Example 1: Determine which of the following matrices are singular. (a) \(\left[\begin{array}{rr}7 & 4 \\ \\ 12 & 4 \end{array}\right]\) (b) \(\left[\begin{array}{rr}2 & 1 & 3 \\ 1 & 0 & 2\\ 6 & 3&1 \end{array}\right]\).
Solution:
We will find the determinants of each of the given matrices.
(a) \(\left\begin{array}{rr}7 & 2 \\ \\ 14 & 4 \end{array}\right\) = 7 × 4  2 × 14 = 28 + 28 = 0.
Thus, the given matrix is a singular matrix.
(b) \(\left\begin{array}{rr}2 & 1 & 3 \\ 1 & 0 & 2\\ 6 & 3&1 \end{array}\right\)
= 2 \(\left\begin{array}{rr}0 & 2 \\ \\ 3 & 1 \end{array}\right\)  (1) \(\left\begin{array}{rr}2 & 3 \\ \\ 6 & 1 \end{array}\right\) + 3 \(\left\begin{array}{rr}1 & 0 \\ \\ 6 & 3 \end{array}\right\)
= 2 (0  6) + 1 (1 + 12) + 3 (3  0)
= 12 + 13 + 9
= 10 ≠ 0
Thus, the given matrix is non singular.
Answer: (a) Singular matrix (b) Nonsingular matrix.

Example 2: Find x if A = \(\left[\begin{array}{rr}x+1 & x & 2 \\ 1 & 0 & 1\\ 4 & 1&x+3 \end{array}\right]\) is a singular matrix.
Solution:
Since A is singular, its determinant is 0. i.e.,
\(\left\begin{array}{rr}x+1 & x & 2 \\ 1 & 0 & 1\\ 4 & 1&x+3 \end{array}\right\) = 0
(x + 1) (0  1)  x (x + 3  4) + 2 (1  0) = 0
x  1  x^{2} + x + 2 = 0
x^{2} + 1 = 0
x^{2} = 1
x = ± 1
Answer: x = 1 or 1 for A to be singular.

Example 3: Determine whether the following system has a unique solution or not: 2x + y + 2z = 3, x + z = 5, 4x + y + 4z = 7.
Solution:
If we write the given system in the matrix form then the corresponding matrix equation is AX = B, then the coefficient matrix is, A = \(\left[\begin{array}{rr}2 & 1 & 2 \\ 1 & 0 & 1\\ 4 & 1&4 \end{array}\right]\).
If the determinant of A is NOT zero (i.e., if A is nonsingular), then only the system has a unique solution (by Cramer's rule)
A = \(\left\begin{array}{rr}2 & 1 & 2 \\ 1 & 0 & 1\\ 4 & 1&4 \end{array}\right\)
= 2 (0  1)  1 (4  4) + 2 (1  0)
= 2 + 0 + 2
= 0
Answer: The system does NOT have a unique solution. It either has an infinite number of solutions or it has no solution.
FAQs on Singular Matrix
What Does a Singular Matrix Mean?
A singular matrix means a square matrix whose determinant is 0 (or) it is a matrix that does NOT have a multiplicative inverse.
How do You Know if a Matrix is Singular Matrix?
We can say that a matrix 'A' is singular if one of the following conditions is satisfied.
 If determinant of A = 0 (or)
 If A is noninvertible.
What is Singular Matrix Formula?
We know that a singular matrix determinant is 0. Thus, the formula for the singular matrix is "A is singular if and only if det(A) = 0".
Why is It Called a Singular Matrix?
The word "singular" means "exceptional" (or) "remarkable". A singular matrix is specifically used to determine whether a matrix has an inverse, rank of a matrix, uniqueness of the solution of a system of equations, etc. It is also used for various purposes in linear algebra and hence the name.
What is a Singular Matrix 3x3?
The determinant of a singular matrix is 0. An example of a 3x3 singular matrix is \(\left[\begin{array}{rr}2 & 1 & 1 \\ 1 & 0 & 1\\ 2 & 1&1 \end{array}\right]\) is singular as its determinant is zero (as its first and third rows are equal).
What Makes a Matrix A a Singular Matrix?
If the determinant of A is 0 then A is singular. If there is no matrix B such that AB = BA = I, then it means that A has no inverse and in this case also, A is said to be singular.
What is the Rank of a Singular Matrix?
If A is a singular matrix of order n, then it means that det A = 0. Then the rank of the matrix is definitely less than the order of the matrix. i.e., rank(A) < n.
What are Singular and Non Singular Matrices?
A singular matrix is a matrix whose determinant is 0 and hence it has no inverse. On the other hand, a nonsingular matrix is a matrix whose determinant is NOT 0 and hence it has an inverse.
What is Singular Matrix Determinant?
A singular matrix has no inverse. We know that the inverse of a matrix A is (adj A)/(det A) and it does NOT exist when det A = 0. Therefore, the determinant of a singular matrix is 0.
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