Singular Matrix

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

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

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

• The plural of Matrixmatrix is matrices.

• The size of a Matrixmatrix is referred to as ‘n by m’ Matrixmatrix 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 Matrixmatrix, that’s because the number of rows here is equal to 3 and the number of columns is equal to 2.

What is the Singular Matrix?

Depending on the determinant, we may tell if a Matrix is Singular or non-Singular. 'det A' or '|A|' denotes the determinant of a Matrix 'A.' When the determinant of a Matrix is zero, it is said to be Singular. 

If the determinant of a Singular Matrix is 0, it is a square Matrix. i.e., if and only if det A = 0, a square Matrix A is Singular.

Since, the inverse of a Matrix A is found using the formula:

A^-1 = (adj A) / (det A). 

And, det A (the determinant of A) is in the denominator and a fraction is NOT defined if its denominator is 0. 

As a result, A-1 is not defined when det A equals 0. i.e., the Singular Matrix’s inverse is not defined. 

Hence, there does not exist any Matrix B such that AB = BA = I (where I is the identity Matrix).

Example of finding Singular Matrix

Two conditions must be met to establish whether a given Matrix is Singular:

• Make sure A is a square Matrix.

• Verify that det A equals 0.

Here are a few examples of how to determine if a Matrix is single.

\[ A = \begin{bmatrix} 3 & 6 \\ 2 & 4 \end{bmatrix}\]

The above equation is a Singular Matrix. 

It’s a square Matrix (of order 2x2) and det A (or) |A| = 3 × 4 - 6 × 2 = 12 - 12 = 0.

Properties

Based on its definition, these are some Singular Matrix properties.

• Singular Matrices are all square Matrices.

• A Singular Matrix's determinant is 0.

• A Singular Matrix is a null Matrix of any order.

• A Singular Matrix's inverse is not specified, making it non-invertible.

• In a Matrix, qualities of determinants

• If any two rows or columns are identical, the determinant is zero, and the Matrix is Singular.

• If all of a row or column's elements are zeros, the determinant is 0 and the Matrix is Singular.

• The determinant is 0 and the Matrix is Singular if one of the rows (columns) is a scalar multiple of the other row (column).

• A Singular Matrix's rank is significantly lower than the Matrix's order. A 3x3 Matrix, for example, has a rank of less than 3.

• A Singular Matrix's rows and columns are not linearly independent.

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

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

For a 2×2 Matrixmatrix - 

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

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

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

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

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

• Step 6- If the value of the determinant (ad-bc = 0), then the Matrixmatrix 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 Matrixmatrix - 

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

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

• Step 3- The determinant of the Matrixmatrix 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 Matrixmatrix.

• 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 Matrixmatrix 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 Matrixmatrix A is said to be non -singular.

How to know if a Matrix is Singular?

According to the singular Matrixmatrix properties,

Questions on singular Matrixmatrix-

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

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

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

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

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

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