Singular Matrix
Last Updated :
18 Feb, 2024
A singular matrix is a square matrix of determinant “0.” i.e., a square matrix A is singular if and only if det A = 0. Inverse of a matrix A is found using the formula A-1 = (adj A) / (det A). Thus, a matrix is called a square matrix if its determinant is zero. Now let us discuss about singular matrix, its properties, and others in detail.
What is a Singular Matrix?
A square matrix is said to be a singular matrix if its determinant is zero and it is not invertible. In a singular matrix, some rows and columns are linearly dependent. As the rows and columns of a singular matrix are linearly dependent, the rank of the matrix will be less than the order of the matrix.
The image given below is an “m × n” matrix that has “m” rows and “n” columns.
We know that the formula to determine the inverse of a matrix is equal to the adjoint of the matrix divided by the determinant of the matrix, i.e., A-1 = (adj A) / |A|. From the definition of a singular matrix, we know that |A| = 0, so its inverse is not defined.
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
- B is said to be the inverse matrix of A
Thus, matrix A is a non-singular matrix.
Properties of a Singular Matrix
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 matrix 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 as its determinant is zero.Â
Differences Between Singular and Non-Singular Matrix
Differences between Singular Matrix and Non-Singular Matrix can be understood using the table given below
Singular Matrix Vs Non-Singular Matrix
|
---|
 Singular MatrixÂ
| Â Non-Singular MatrixÂ
|
---|
A square matrix is said to be a singular matrix if its determinant is zero, i.e., det A = 0.
| A square matrix is said to be a non-singular matrix if its determinant is not zero, i.e., det A ≠0.
|
If a matrix is singular, then its inverse is not defined.
| If a matrix is non-singular, then its inverse is defined.
|
The rank of a singular matrix will be less than the order of the matrix, i.e., Rank (A) < Order of A.
| The rank of a non-singular matrix will be equal to the order of the matrix, i.e., Rank (A) = Order of A.
|
In a singular matrix, some rows and columns are linearly dependent.
| In a non-singular matrix, all the rows and columns are linearly independent.
|
[Tex]A = \left(\begin{array}{ccc} 2 & 2 & 4\\ 1 & 1 & 2\\ 3 & 7 & 9 \end{array}\right)
[/Tex]
| [Tex]B = \left[\begin{array}{ccc} 1 & 2 & -3\\ 6 & 0 & 8\\ -1 & 4 & 0 \end{array}\right]
[/Tex]
|
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 a square matrix or not.
- If the given matrix is a square matrix, 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.
Formula for Determinant of “2 × 2” Matrix
If A =Â [Tex]\left[\begin{array}{cc} a & b\\ c & d \end{array}\right]Â Â Â Â
[/Tex] is a “2 × 2” matrix, then its determinant isÂ
|A|= [ad – bc]
Formula for Determinant of “3 × 3” Matrix
If A =Â [Tex]\left[\begin{array}{ccc} a_{1} & a_{2} & a_{3}\\ b_{1} & b_{2} & b_{3}\\ c_{1} & c_{2} & c_{3} \end{array}\right]Â Â Â Â
[/Tex] is a “3 × 3” matrix, then its determinant isÂ
|A|= a1(b2c3 – b3c2) – a2(b1c3 – b3c1) + a3(b1c2 – b2c1)
Also, Check
Solved Examples on Singular Matrix
Example 1: Find the value of k if the matrix given below, is a singular matrix.
[Tex]A = \left[\begin{array}{cc} k & -4\\ 5 & 2 \end{array}\right]
[/Tex]
Solution:
Given matrix A =Â [Tex]\left[\begin{array}{cc} k & -4\\ 5 & 2 \end{array}\right]
[/Tex]
We know that the determinant of a singular matrix is zero, i.e., det A = 0
⇒ (2×k) – (–4 × 5) = 0
⇒ 2k + 20 = 0
⇒ 2k = -20
⇒ k = –20/2 = –10
Hence, the value of k if the given matrix is a singular matrix is –10.
Example 2: Determine the inverse of the matrix given below.
[Tex]P = \left[\begin{array}{cc} -3 & 4\\ 6 & -8 \end{array}\right]
[/Tex]
Solution:
Given matrix [Tex] P = \left[\begin{array}{cc} -3 & 4\\ 6 & -8 \end{array}\right]
[/Tex]
P-1 = Adj P / |P|
Now, let us find the determinant of the matrix P.
|P| = (–3 × –8) – (6 × 4)
|P| = 24 – 24 = 0
Since, the determinant of matrix P = 0, it is a singular matrix, and its inverse matrix doesn’t exist.
Example 3: Determine whether the given matrix is singular or not.
[Tex]A = \left[\begin{array}{ccc} 1 & 0 & -3\\ 0 & 5 & 2\\ -1 & 4 & 0 \end{array}\right]
[/Tex]
Solution:
Given matrix A =Â [Tex]\left[\begin{array}{ccc} 1 & 0 & -3\\ 0 & 5 & 2\\ -1 & 4 & 0 \end{array}\right]
[/Tex]
To determine whether the given matrix is singular or not, we have to find its determinant.
det A = 1[(5 × 0) – (4 × 2)] – 0[(0 × 0) – (2 × –1)] + (-3) [(0 × 4) – (–1 × 5)]
⇒ |A| = (1 × -8) – 0 + (–3 × 5)Â
⇒ |A| = –8 – 15 = –23 ≠0
Since the determinant of the given matrix is not equal to zero, it is a non-singular matrix.
Example 4: Find the value of b if the matrix given below, is a singular matrix.
[Tex]B = \left[\begin{array}{cc} 9 & b\\ 6 & -2 \end{array}\right]
[/Tex]
Solution:
Given matrix [Tex]B = \left[\begin{array}{cc} 9 & b\\ 6 & -2 \end{array}\right]
[/Tex]
We know that the determinant of a singular matrix is zero, i.e., det B = 0
⇒ (9 × –2) – (6 × b) = 0
⇒ –18 – 6b = 0
⇒ –6b = 18
⇒ b = 18/–6 = –3
Hence, the value of b if the given matrix is a singular matrix is –3.
FAQs on Singular Matrix
1. Define a Matrix.
A matrix is defined as a rectangular array of numbers that are arranged in rows and columns.
2. What is a Singular Matrix?
A square matrix is said to be a singular matrix if its determinant is zero and it is not invertible.
3. What is the Rank of a Singular Matrix of Order “3 × 3”?
If the given matrix A is singular, then its determinant is zero. Now, the rank of the given matrix will be less than the order of the matrix, i.e., rank (A) < 3.
4. What is the Determinant of a Singular Matrix?
The determinant of a matrix determines whether it is singular or non-singular. So, a matrix is said to be singular if its determinant is zero.
5. Is a Zero Matrix a Singular Matrix?
As the determinant of a singular matrix is zero, it is a singular matrix.
6. What is Rank of a Singular Matrix?
The rank of singular matrix ‘n’ is always less than ‘n’.
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