What Is Non Singular Matrix?

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What Is Non Singular Matrix?

In the realm of mathematics, matrices play a crucial role in various fields, including algebra, computer science, and engineering. One fundamental concept within this domain is the notion of non-singular matrices. In this article, we’ll delve into what non-singular matrices are, their properties, and provide illustrative examples to facilitate understanding.

What Is A Matrix?

Before diving into non-singular matrices, let’s briefly review what a matrix is. In simple terms, a matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. For instance, consider the following matrix:

A= ⎡1 2 3⎤

       4 5 6

     ⎣7 8 9⎦

​Here, A is a 3×3 matrix consisting of three rows and three columns.

Understanding Singularity

In the context of matrices, singularity refers to a special property where the matrix fails to have an inverse. In other words, a singular matrix is non-invertible, making it impossible to find a unique solution to a system of equations represented by that matrix.

What Is A Non-Singular Matrix?

Conversely, a non-singular matrix, also known as an invertible or regular matrix, possesses an inverse. This means that for every non-singular matrix, there exists another matrix, known as its inverse, such that when the two are multiplied together, the result is the identity matrix.

Properties Of Non-Singular Matrices

Non-singular matrices exhibit several key properties:

  • Existence of Inverse: As mentioned earlier, every non-singular matrix has an inverse.
  • Determinant Non-Zero: The determinant of a non-singular matrix is non-zero. This property is crucial because the determinant is a factor in determining whether a matrix is singular or non-singular.
  • Unique Solution: Systems of linear equations represented by non-singular matrices have unique solutions. This property makes non-singular matrices particularly useful in solving equations and modeling various real-world scenarios.

Non-Singular Matrix Formula

The condition for a square matrix A to be non-singular is that its determinant (∣A∣) must be non-zero. Mathematically, this can be expressed as:


Examples Of Non-Singular Matrices

Let’s explore some examples to better grasp non-singular matrices:

2×2 Non-Singular Matrix Example:

B= ⎡2 1⎤

      ⎣4 3⎦

3×3 Non-Singular Matrix Example:

C= ⎡1 0 2⎤

       0 1 3

      ⎣0 0 1⎦


Non-singular matrices find applications in various fields, including:

  • Engineering: Used in solving systems of linear equations representing physical phenomena.
  • Computer Graphics: Essential for transformations such as rotations and translations.
  • Statistics: Employed in multivariate analysis and regression models.


In summary, non-singular matrices are fundamental in linear algebra and have widespread applications across diverse disciplines. Understanding their properties and significance enables mathematicians, scientists, and engineers to solve complex problems efficiently. By grasping the concepts outlined in this article, you’ll be better equipped to navigate the realm of matrices and their applications in the real world.


What Is The Meaning Of Non-Singular Matrix?

A non-singular matrix is a square one whose determinant is not zero. The rank of a matrix [A] is equal to the order of the largest non-singular submatrix of [A]. It follows that a non-singular square matrix of n × n has a rank of n.

How Do You Make A Matrix Non Singular?

Adding a tiny bit of noise to a singular matrix makes it non-singular.

What Is Non-Singular Matrix Class 12?

Non Singular matrix is a square matrix whose determinant is a non-zero value. The non-singular matrix property is to be satisfied to find the inverse of a matrix. For a square matrix A = [abcd] [ a b c d ] , the condition of it being a non singular matrix is the determinant of this matrix A is a non zero value.

What Is A Singular Matrix?

The matrices are known to be singular if their determinant is equal to the zero. For example, if we take a matrix x, whose elements of the first column are zero. Then by the rules and property of determinants, one can say that the determinant, in this case, is zero. Therefore, matrix x is definitely a singular matrix.

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