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".
What is 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 = \begin{bmatrix}
5&6& 0\\
4& 2 & 3\\
1 & 10& 9
\end{bmatrix} is a non-singular matrix or not?
Solution:
First, we find determinant of C i.e., |C| = \begin{vmatrix}
5&6& 0\\
4& 2 & 3\\
1 & 10& 9
\end{vmatrix}
|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 = \begin{bmatrix}
10 & 7\\
4 & 2
\end{bmatrix} is singular or non-singular?
Solution:
First, we find the determinant of A i.e., |A| = \begin{bmatrix}
10 & 7\\
4 & 2
\end{bmatrix}
|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.
Properties of 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.
How to Identify Non-Singular Matrix
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.
Difference Between Singular and Non-Singular Matrix
The below table represents the difference between singular and non-singular matrices.
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. |
|---|
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Solved Examples on Non-Singular Matrix
Example 1: Check whether the given matrix A = \begin{bmatrix}
2 & 0\\
5 & 9
\end{bmatrix} is a non-singular matrix or not?
Solution:
First, we find the determinant of A i.e., |A| = \begin{vmatrix}
2 & 0\\
5 & 9
\end{vmatrix}
|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 = \begin{bmatrix}
2 & 1\\
8 & 4
\end{bmatrix} is a non-singular matrix or not?
Solution:
First, we find the determinant of B i.e., |B| = \begin{vmatrix}
2 & 1\\
8 & 4
\end{vmatrix}
|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 = \begin{bmatrix}
1 & 5 & 3\\
0 & 2& 1\\
7 & 9 & 4
\end{bmatrix} is singular or non-singular?
Solution:
First, we find determinant of P i.e., |P| = \begin{vmatrix}
1 & 5 & 3\\
0 & 2& 1\\
7 & 9 & 4
\end{vmatrix}
|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 = \begin{bmatrix}
5 & 0 & -2\\
1 & 3& 2\\
2 & 6 & 4
\end{bmatrix} is singular or non-singular?
Solution:
First, we find determinant of Q i.e., |Q| = \begin{vmatrix}
5 & 0 & -2\\
1 & 3& 2\\
2 & 6 & 4
\end{vmatrix}
|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 = \begin{bmatrix}
2 & 7 & 12\\
4 & 6& 1\\
3 & 0 & 5
\end{bmatrix} is a non-singular matrix or not?
Q2. Determine the matrix P = \begin{bmatrix}
0 & 4\\
7&1
\end{bmatrix} is singular or non-singular?
Q3. Check whether the given matrix A = \begin{bmatrix}
2 & 1 & 3\\
6 & 1& 1\\
-24 & -2 & 4
\end{bmatrix} is a non-singular matrix or not?
Q4. Determine the matrix P = \begin{bmatrix}
2 & 3\\
6& 9
\end{bmatrix} is singular or non-singular?