Table of Content

• What is Non-Singular Matrix?
• Properties of Non-Singular Matrix
• How to Identify Non-Singular Matrix
• Difference Between Singular and Non-Singular Matrix
• Solved Examples on Non-Singular Matrix

Characteristics	Singular Matrix	Non-Singular Matrix
Definition	Singular matrix is a matrix whose determinant is zero.	Non-singular matrix is a matrix whose determinant is non-zero.
Condition	|A| = 0 then, A is singular matrix.	|A| ≠ 0 then, A is non-singular matrix.
Invertible	Singular matrices are not invertible.	Non-singular matrices are invertible.
Examples	Null or Zero matrix is an example of singular matrix.	Identity matrix is an example of non-singular matrix.

Read More About,

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

Solved Examples on Non-Singular Matrix

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.

Practice Questions on 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?

FAQs on Non-Singular Matrix

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.