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
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 non-invertible.
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 non-singular 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.
FAQs
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Founded in 2013 by Manan Khurma, Cuemath is an ed tech company focused on math learning for K-12 students. Our mission is to make the world's children great at math, and create invincible problem-solvers who will go on to solve humanity's greatest problems.
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To become a tutor, you should get your A Levels in the subject you want to teach from either a sixth form or a college. After that you have a few options: a maths degree is the obvious choice. However, there are other related undergraduate degrees that also will provide you with the skills needed to teach maths.
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U.S. ranks near the bottom
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