Non Singular Matrix

A non-singular matrix (also called an invertible matrix) is a square matrix whose determinant is a non-zero value. It is used to find the inverse of a matrix.

The condition for a matrix to be non-singular:

• Determinant of the matrix should be non-zero: det(A) ≠ 0.
• Has an inverse: There exists a unique inverse matrix A⁻¹.
• A non-singular matrix has rank equal to its order (i.e., rank = number of rows = number of columns).

Non-Singular Matrix Example

Some examples of non-singular matrices are:

Example: Check if 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.

Singular vs Non-Singular Matrix

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

Also Check

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

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?