Non Singular Matrix

Non-singular matrix is a square whose determinant is not zero. The non-singular matrices are also invertible matrices. In this article we will explore non-singular matrix in detail along with the non-singular matrix definition, non-singular matrix examples. We will also discuss how to find a matrix is non-singular or not, properties of non-singular matrix and solve some examples related to non-singular matrix. Let’s start our learning on the topic “Non-Singular Matrix”.

A non-singular matrix is a matrix with non-zero determinant. The matrices whose determinant is not equal to zero are known as non-singular matrices. The condition for a matrix to be non-singular is that the determinant of the matrix should be non-zero. The condition for a non-singular matrix can be mathematically represented as Det (Matrix) ≠ 0 or |Matrix| ≠ 0. The singular matrices have an inverse, so they are also called invertible matrices.

Non-Singular Matrix Definition

A square matrix whose determinant is non-zero is referred to as non-singular matrix. In other words, a square matrix with its determinant not equal to zero is called as non-singular matrix.

If |A| ≠ 0 then, A is non-singular matrix

Non-Singular Matrix Example

Some examples of non-singular matrix are:

Example: Check the matrix C = [Tex]\begin{bmatrix} 5&6& 0\\ 4& 2 & 3\\ 1 & 10& 9 \end{bmatrix}[/Tex] is a non-singular matrix or not?

Solution:

First, we find determinant of C i.e., |C| = [Tex]\begin{vmatrix} 5&6& 0\\ 4& 2 & 3\\ 1 & 10& 9 \end{vmatrix}[/Tex]

|C| = 5 × [(2 × 9) – (3 × 10)] – 6 × [(9 × 4) – (3 × 1)] + 0 × [(4 × 10) – (2 × 1)]

|C| = 5 × [18 – 30] – 6 × [36 – 3] + 0

|C| = 5 × (-12) – 6 × (33)

|C| = -60 – 198

|C| = -258

Since, |C| is not equal to zero the given matrix C is a non-singular matrix.

Example: Check whether the matrix A = [Tex]\begin{bmatrix} 10 & 7\\ 4 & 2 \end{bmatrix}[/Tex] is singular or non-singular?

Solution:

First, we find the determinant of A i.e., |A| = [Tex]\begin{bmatrix} 10 & 7\\ 4 & 2 \end{bmatrix}[/Tex]

|A| = (2 × 10) – (7 × 4)

|A| = 20 – 28

|A| = -8

Since, |A| is not equal to zero the given matrix A is non-singular matrix.

Some properties of non-singular matrix are listed below.

• The determinant is a non-zero value for the non-singular matrix.
• Non-singular matrix is a square matrix.
• Non-singular matrices are invertible as its determinant is not equal to zero.
• The multiplication of two non-singular matrices is also non-singular matrix.
• A matrix kP is non-singular matrix if P is non-singular matrix and k is constant.

The below are some steps to find the matrix is non-singular matrix or not.

• First, find the determinant of the given matrix.
• If the determinant is zero, the matrix is singular matrix.
• If the determinant is non-zero then, the matrix is non-singular matrix.

The below table represents the difference between singular and non-singular matrices.

Read More About,

• Inverse of a Matrix
• Singular Matrix
• Types of Matrices

Example 1: Check whether the given matrix A = [Tex]\begin{bmatrix} 2 & 0\\ 5 & 9 \end{bmatrix}[/Tex] is a non-singular matrix or not?

Solution:

First, we find the determinant of A i.e., |A| = [Tex]\begin{vmatrix} 2 & 0\\ 5 & 9 \end{vmatrix}[/Tex]

|A| = (2 × 9) – (0 × 5)

|A| = 18 – 0

|A| = 18

Since, |A| is not equal to zero the given matrix A is non-singular matrix.

Example 2: Find whether the given matrix B = [Tex]\begin{bmatrix} 2 & 1\\ 8 & 4 \end{bmatrix}[/Tex] is a non-singular matrix or not?

Solution:

First, we find the determinant of B i.e., |B| = [Tex]\begin{vmatrix} 2 & 1\\ 8 & 4 \end{vmatrix}[/Tex]

|B| = (2 × 4) – (1 × 8)

|B| = 8 – 8

|B| = 0

Since, |B| is equal to zero the given matrix B is not a non-singular matrix.

Example 3: Determine the matrix P = [Tex]\begin{bmatrix} 1 & 5 & 3\\ 0 & 2& 1\\ 7 & 9 & 4 \end{bmatrix}[/Tex] is singular or non-singular?

Solution:

First, we find determinant of P i.e., |P| = [Tex]\begin{vmatrix} 1 & 5 & 3\\ 0 & 2& 1\\ 7 & 9 & 4 \end{vmatrix}[/Tex]

|P| = 1 × [(2 × 4) – (9 × 1)] – 5 × [(0 × 4) – (7 × 1)] + 3 × [(0 × 9) – (7 × 2)]

|P| = 1 × [8 – 9] – 5 × [0 – 7] + 3 × [0 – 14]

|P| = 1 × (-1) – 5 × (- 7) + 3 × (- 14)

|P| = -1 + 35 – 42

|P| = -7

Since, |P| is not equal to zero the given matrix P is a non-singular matrix.

Example 4: Determine the matrix Q = [Tex]\begin{bmatrix} 5 & 0 & -2\\ 1 & 3& 2\\ 2 & 6 & 4 \end{bmatrix}[/Tex] is singular or non-singular?

Solution:

First, we find determinant of Q i.e., |Q| = [Tex]\begin{vmatrix} 5 & 0 & -2\\ 1 & 3& 2\\ 2 & 6 & 4 \end{vmatrix}[/Tex]

|Q| = 5 × [(3 × 4) – (6 × 2)] – 0 × [(1 × 4) – (2 × 2)] + (-2) × [(1 × 6) – (3 × 2)]

|Q| = 5 × [12 – 12] – 0 × [4 – 4] + (-2) × [6 – 6]

|Q| = 5 × 0 – 0 – 2 × 0

|Q| = 0

Since, |Q| is equal to zero the given matrix Q is not a non-singular matrix.

Q1. Check whether the given matrix A = [Tex]\begin{bmatrix} 2 & 7 & 12\\ 4 & 6& 1\\ 3 & 0 & 5 \end{bmatrix}[/Tex] is a non-singular matrix or not?

Q2. Determine the matrix P = [Tex]\begin{bmatrix} 0 & 4\\ 7&1 \end{bmatrix}[/Tex] is singular or non-singular?

Q3. Check whether the given matrix A = [Tex]\begin{bmatrix} 2 & 1 & 3\\ 6 & 1& 1\\ -24 & -2 & 4 \end{bmatrix}[/Tex] is a non-singular matrix or not?

Q4. Determine the matrix P = [Tex]\begin{bmatrix} 2 & 3\\ 6& 9 \end{bmatrix}[/Tex] is singular or non-singular?

What is a 2×2 Non-Singular Matrix?

A 2×2 non-singular matrix is a 2×2 matrix with non-zero determinant.

How Do You if a Matrix is Singular or Not?

To know if a matrix is singular or non-singular we determine the determinant of the matrix. If determinant is zero matrix is singular otherwise the matrix is non-singular.

Is Identity Matrix a Non-Singular Matrix?

Yes, an identity matrix is a non-singular matrix as its determinant is non-zero.

What is Difference Between Singular and Non-singular Matrix?

The singular matrix is a matrix with its determinant zero whereas the non-singular matrix is a matrix with its determinant non-zero.

What is the Condition for Non-Singular Matrix?

The condition for non-singular matrix is |Matrix| ≠ 0.