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.

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

Here, m = The number of rows

n= The number of columns

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

Singular matrix example-

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.

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.

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.

According to the singular matrix properties,

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)