Singular Matrix

What is a matrix?

  • A matrix is a rectangular array of numbers or symbols which are generally arranged in rows and columns. 

  • The order of the matrix is defined as the number of rows and columns.

  • The entries are the numbers in the matrix and each number is known as an element.

  • The plural of matrix is matrices.

  • The size of a matrix is referred to as ‘n by m’ matrix and is written as n×m where n is the number of rows and m is the number of columns.

  • For example, we have a 3×2 matrix, that’s because the number of rows here is equal to 3 and the number of columns is equal to 2.                               


Let’s first know what a Square Matrix is!

Square matrix is a matrix where the number of columns is equal to the number of rows.

If m=n, the matrix is a square matrix

If m ≠n, the matrix is a rectangular matrix


                  Here, m = The number of rows

                           n= The number of columns

What is a Singular Matrix?

A matrix is said to be singular if and only if its determinant is equal to zero.

Singular matrix example-

 

Singular Matrix Properties-

The singular matrix properties are listed below:

  • A matrix is said to be singular if and only if its determinant is equal to zero.

  • A singular matrix is a matrix that has no inverse such that it has no multiplicative inverse.

Necessary Condition for Existence of the inverse of a Matrix –

For any square matrix A, A should not be singular (|A| ≠0), which means that the determinant of the matrix should not be equal to zero.

Steps to find the determinant (d) of a matrix-

Before, we know how to check whether a matrix is singular or not, we need to know how to calculate the determinant of a matrix.

For a 2×2 matrix - 

Step 1 – First of all check whether the matrix is a square matrix or not.

Step 2- For a 2×2 matrix (2 rows and 2 columns),

                           

                   

 Step 3- The determinant of the matrix A = ad-bc, and is represented by |A|

 Step 4 – The determinant of matrix A = a times d minus b times c.

 Step 5 - If the value of the determinant (ad-bc = 0), then the matrix A is said to be singular.

Step 6 - If the value of the determinant (ad-bc = 0), then the matrix A is said to be non- singular.

An easy way to remember – 


           

Here, Blue is positive (+ve) = (+ad),


         Red is negative (-ve) = (-bc)


Here’s an example for better understanding,

We know that, to calculate the determinant, 

|A| = 2×5 - 2×4 

= 10- 8 = 2


For a 3×3 matrix - 

Step 1 – First of all check whether the matrix is a square matrix or not.

Step 2- For a 3×3 matrix (3 rows and 3 columns),

                    

               

 Step 3- The determinant of the matrix A = a1(b2c3 – b3c2) - a2(b1c3 – b3c1) – a3(b1c2 – b2c1), and is represented by |A|

 Step 4 – Multiply a1 by the determinant of the 2×2 matrix.

 Step 5 – Likewise do it for a2 and a3

 Step 6 – Sum all of them, do not forget the minus signs before 

 Step 7 - If the value of the determinant (a1(b2c3 – b3c2) - a2(b1c3 – b3c1) – a3(b1c2 – b2c1) = 0), then the matrix A is said to be singular. 

Step 8 - If the value of the determinant (a1(b2c3 – b3c2) - a2(b1c3 – b3c1) – a3(b1c2 – b2c1) ≠ 0), then the matrix A is said to be non -singular.

How to know if a Matrix is Singular?

According to the singular matrix properties,

For a 2×2 matrix,

                    

               

The determinant of the matrix A = ad-bc,

If the value of the determinant (ad-bc = 0) 


For a 3×3 matrix,



The determinant of A is equal to zero.

  

 A= a1(b2c3 – b3c2) - a2(b1c3 – b3c1) – a3(b1c2 – b2c1) = 0


Questions on singular matrix-

Question 1) Find the inverse of the given matrix below.

           

Solution) Since the above matrix is a 2×2 matrix,

Comparing the matrix with the general form,

Here, the value of a = 2, b = 4, c= 2 and d = 4.

Then, determinant of A (|A|) = ad-bc

(2×4 - 4×2 = 0)

According to the singular matrix definition we know that the determinant needs to be zero. Since the determinant of the matrix A = 0, it is a singular matrix and has no inverse.


Question 2) Find whether the given matrix is singular or not.

     

Solution) Since the above matrix is a 2×2 matrix,

Comparing the matrix with the general form,

Here, the value of a = 8, b = 7, c= 4 and d = 5.

Then, determinant of A (|A|) = ad-bc

(8×5 - 7×4 = 12) 

According to the singular matrix definition we know that the determinant needs to be zero. Since the determinant of the matrix A = 12, it is not a singular matrix.


FAQ (Frequently Asked Questions)

1. How do you know if a matrix is singular?

According to the singular matrix properties, a square matrix is said to be singular if and only if the determinant of the matrix is equal to zero.

2. What is a singular matrix?

According to the singular matrix definition, when a matrix is said to be singular it means that the matrix is non-invertible. In a singular matrix, the determinant is always equal to zero.

3. Does a singular matrix have a solution?

There is a solution set which has an infinite number of solutions if the system has a singular matrix.

4. Define the singular matrix and non-singular matrix? Give a singular matrix example and non-singular matrix example.

 Let’s define  singular matrix and a non- singular matrix.

 If a matrix A does not have an inverse then it is said to be a singular matrix. A matrix B such that AB = BA = identity matrix (I) is known as the inverse of matrix A. A non – singular matrix is a square matrix which has a matrix inverse. In simpler words, a non-singular matrix is one which is not singular. If the determinant of a matrix is not equal to zero then it is known as a non-singular matrix.

Singular matrix example – 

                                                    

 is a singular matrix,

Since the determinant of the above matrix is = (2×1 - 1×2 = 0) 

Non-singular matrix example - 



is a non-singular matrix.

Since the determinant of the above matrix is = (3×2- 2×1 = 4)