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# Singular Matrix

• Last Updated : 09 Jan, 2023

A matrix is defined as a rectangular array of numbers that are arranged in rows and columns. The size of a matrix can be determined by the number of rows and columns in it. A matrix is said to be an “m by n” matrix when it has “m” rows and “n” columns and is written as an “m Ã— n” matrix. For example, the order of the matrix that has five rows and four columns is “5 Ã— 4.” We have different types of matrices, such as rectangular matrices, square matrices, triangular matrices, symmetric matrices, singular matrices, etc. The determinant of a matrix determines whether it is singular or non-singular. Now let us discuss a singular matrix in detail. The image given below is an “m Ã— n” matrix that has “m” rows and “n” columns.

## What is a Singular Matrix?

A square matrix is said to be a singular matrix if its determinant is zero and it is not invertible. In a singular matrix, some rows and columns are linearly dependent. As the rows and columns of a singular matrix are linearly dependent, the rank of the matrix will be less than the order of the matrix.

We know that the formula to determine the inverse of a matrix is equal to the adjoint of the matrix divided by the determinant of the matrix, i.e., A-1 = (adj A) / |A|. From the definition of a singular matrix, we know that |A| = 0, so its inverse is not defined.

Let us consider that A and B are two square matrices of order “n Ã— n”

If,

AB = BA = I

where I is an identity or unit matrix of order n, then matrix B is said to be the inverse matrix of A. Thus, matrix A is a non-singular matrix.

## Properties of a Singular Matrix

The following are the properties of the Singular Matrix:

• Every singular matrix must be a square matrix, i.e., a matrix that has an equal number of rows and columns.
• The determinant of a singular matrix is equal to zero.
• As the determinant of a singular matrix is zero, its inverse is not defined.
• A zero matrix of any order matrix is a singular matrix, as its determinant is zero.
• In a singular matrix, some rows and columns are linearly dependent.
• The rank of a singular matrix will be less than the order of the matrix, i.e., Rank (A) < Order of A.
• A matrix that has any two rows or any two columns identical is singular, as the determinant of such a matrix is zero.
• When a row or column’s elements in a matrix are all zeros, then the matrix is singular, as its determinant is zero.
• When one row (or column) of a matrix is a scalar multiple of another row (or column), then the matrix is singular as its determinant is zero.

## Differences between Singular and Non-Singular Matrix

Differences between Singular Matrix and Non-Singular Matrix can be understood using the table given below

## Identifying a Singular Matrix

Follow the conditions given below to determine whether the given matrix is singular or not.

• Determine whether the given matrix is a square matrix or not.
• If the given matrix is a square matrix, then find the determinant of the matrix.

â‡’ If |A|= 0, then the given matrix is singular.

â‡’ If |A|â‰ 0, then the given matrix is non-singular.

### Formula for Determinant of “2 Ã— 2” Matrix

If A =  is a “2 Ã— 2” matrix, then its determinant is

### Formula for Determinant of “3 Ã— 3” Matrix

If A =  is a “3 Ã— 3” matrix, then its determinant is

|A|= a1(b2c3 â€“ b3c2) â€“ a2(b1c3 â€“ b3c1) + a3(b1c2 â€“ b2c1)

Also, Check

## Solved Examples on Singular Matrix

Example 1: Find the value of k if the matrix given below, is a singular matrix.

Solution:

Given matrix A =

We know that the determinant of a singular matrix is zero, i.e., det A = 0

â‡’ (2Ã—k) â€“ (â€“4 Ã— 5) = 0

â‡’ 2k + 20 = 0

â‡’ 2k = -20

â‡’ k = â€“20/2 = â€“10

Hence, the value of k if the given matrix is a singular matrix is â€“10.

Example 2: Determine the inverse of the matrix given below.

Solution:

Given matrix

P-1 = Adj P / |P|

Now, let us find the determinant of the matrix P.

|P| = (â€“3 Ã— â€“8) â€“ (6 Ã— 4)

|P| = 24 â€“ 24 = 0

Since, the determinant of matrix P = 0, it is a singular matrix, and its inverse matrix doesn’t exist.

Example 3: Determine whether the given matrix is singular or not.

Solution:

Given matrix A =

To determine whether the given matrix is singular or not, we have to find its determinant.

det A = 1[(5 Ã— 0) â€“ (4 Ã— 2)] â€“ 0[(0 Ã— 0) â€“ (2 Ã— â€“1)] + (-3) [(0 Ã— 4) â€“ (â€“1 Ã— 5)]

â‡’ |A| = (1 Ã— -8) â€“ 0 + (â€“3 Ã— 5)

â‡’ |A| = â€“8 â€“ 15 = â€“23 â‰  0

Since the determinant of the given matrix is not equal to zero, it is a non-singular matrix.

Example 4: Find the value of b if the matrix given below, is a singular matrix.

Solution:

Given matrix

We know that the determinant of a singular matrix is zero, i.e., det B = 0

â‡’ (9 Ã— â€“2) â€“ (6 Ã— b) = 0

â‡’ â€“18 â€“ 6b = 0

â‡’ â€“6b = 18

â‡’ b = 18/â€“6 = â€“3

Hence, the value of b if the given matrix is a singular matrix is â€“3.

## FAQs on Singular Matrix

### Question 1: Define a matrix.

A matrix is defined as a rectangular array of numbers that are arranged in rows and columns.

### Question 2: What is a singular matrix?

A square matrix is said to be a singular matrix if its determinant is zero and it is not invertible.

### Question 3: What is the rank of a singular matrix of order “3 Ã— 3”?

If the given matrix A is singular, then its determinant is zero. Now, the rank of the given matrix will be less than the order of the matrix, i.e., rank (A) < 3.

### Question 4: What is the determinant of a singular matrix?

The determinant of a matrix determines whether it is singular or non-singular. So, a matrix is said to be singular if its determinant is zero.