Singular Matrix: Definition, Properties and Examples

Q: Ques 1- How do you know if a matrix is singular?

Ans- If this matrix is singular, i.e., it has determinant zero (0), this corresponds to the parallelepiped being wholly flattened, a line, or just a point. You can think of a×a standard matrix as a linear transformation.

Q: Ques 2- Does a singular matrix have a solution?

Ans- Equations including the non-singular matrix have one and only one solution, but equations including a singular matrix are more complex. r = Ax − b.

Q: Ques 3- Are singular matrices invertible?

Ans- Non-square matrices (m-by-n matrices where m ≠ n) do not have an inverse. But, in some cases, such a matrix may have a leftward inverse or right inverse. A square matrix that is non-invertible is known as singular or degenerate. A square matrix is singular if and only if the value of the determinant is 0.

Singular Matrix: A matrix is a set of rectangular arrays arranged in an ordered way, each containing a function or numerical value enclosed in square brackets. Each row and column have values or expressions known as elements or entries. Size or dimension is determined by the total number of rows over the number of columns. In this article, we will explain the types of matrix and properties of singular matrix and determinant by using step-wise examples.

Singular Matrix: Definition

A square matrix, which is non-invertible, is known as singular or degenerate. One can say that if a determinant of a square matrix is zero, it is singular.

If we suppose that,

P and Q are two (2) matrices of the order, a x a satisfying the below condition-

PQ = I = QP

Where ‘I’ represents the ‘Identity matrix’ whose order is ‘a’.

Then, matrix Q is called the inverse of matrix P.

Therefore, P is called a non-singular matrix.

What is Determinant?

Matrixes are arrays of many numbers. With a square matrix, i.e., a matrix with the same rows and columns, important information can be captured in a single number, which is called a determinant. All square matrix has a determinant. It is a mathematical concept that has an essential role in finding the solution and analysis of linear equations.

Suppose, ‘M’ is a matrix as given below:

M = [a b c d e f g h i]

The determinant of the matrix M is represented |M|, such that-

| M | = | a b c d e f g h i |

The determinant can be evaluated as-

| M | = a ( e i – f h ) – b ( d i – g f ) + c ( d h – e g )

To determine a Singular matrix, the value of the determinant has to be equal to 0, i.e. |M| = 0.

So, a(ei – fh) – b(di – fg) + c(dh – eg) = 0.

Properties of Singular Matrix

Here are some important properties of a singular matrix mentioned in the following points:

The value of the determinant of a singular matrix is zero (0).
A non-invertible matrix is introduced as a singular matrix, i.e., when the value determinant of a matrix is zero, we cannot get its inverse.
A singular matrix is described only for square matrices.
There is no multiplicative inverse for this matrix.

FAQs About Singular Matrix

Below are the frequently asked questions about Singular Matrix:

Q1. How do you know if a Matrix is singular?
A. If the Matrix is singular, i.e., it has determinant zero (0) and corresponds to the parallelepiped being wholly flattened, a line, or just a point. You can think of a×a standard matrix as a linear transformation.

Q2. Does a singular matrix have a solution?
A. Equations including the non-singular matrix have one and only one solution, but equations including a singular matrix are more complex. r = Ax − b.

Q3. Are singular matrices invertible?
A. Non-square matrices (m-by-n matrices where m ≠ n) do not have an inverse. But, in some cases, such a matrix may have a leftward inverse or right inverse. A square matrix that is non-invertible is known as singular or degenerate. A square matrix is singular if and only if the value of the determinant is 0.

Q4. What is non-singular matrix?
A. A Non-Singular matrix is a square matrix whose determinant is a non-zero value.

Q5. What is a singular matrix?
A. A matrix is singular if its determinant is 0.

In conclusion, Singular matrices function as a boundary within matrices whose determinants are positive and the matrices whose determinants are negative. The symbol of the determinant has implications in many fields.

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