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A Matrixmatrix is a rectangular array of numbers or symbols which are generally arranged in rows and columns.
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The order of the Matrixmatrix is defined as the number of rows and columns.
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The entries are the numbers in the Matrixmatrix and each number is known as an element.
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The plural of Matrixmatrix is matrices.
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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.
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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.
Contents
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:
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 2×2) and det A (or) |A| = 3 × 4 – 6 × 2 = 12 – 12 = 0.
Properties
Based on its definition, these are some Singular Matrix properties.
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Singular Matrices are all square Matrices.
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A Singular Matrix’s determinant is 0.
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A Singular Matrix is a null Matrix of any order.
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A Singular Matrix’s inverse is not specified, making it non-invertible.
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In a Matrix, qualities of determinants
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If any two rows or columns are identical, the determinant is zero, and the Matrix is Singular.
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If all of a row or column’s elements are zeros, the determinant is 0 and the Matrix is Singular.
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The determinant is 0 and the Matrix is Singular if one of the rows (columns) is a scalar multiple of the other row (column).
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A Singular Matrix’s rank is significantly lower than the Matrix’s order. A 3×3 Matrix, for example, has a rank of less than 3.
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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 –
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Step 1– First of all check whether the Matrixmatrix is a square Matrixmatrix or not.
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Step 2- For a 2×2 Matrixmatrix (2 rows and 2 columns),
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Step 3- The determinant of the Matrixmatrix A = ad-bc, and is represented by |A|
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Step 4 – The determinant of Matrixmatrix A = a times d minus b times c.
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Step 5- If the value of the determinant (ad-bc = 0), then the Matrixmatrix A is said to be singular.
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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 –
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Step 1 – First of all check whether the Matrixmatrix is a square Matrixmatrix or not.
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Step 2- For a 3×3 Matrixmatrix (3 rows and 3 columns),
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Step 3- The determinant of the Matrixmatrix A = a1(b2c3 – b3c2) – a2(b1c3 – b3c1) – a3(b1c2 – b2c1), and is represented by |A|
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Step 4 – Multiply a1 by the determinant of the 2×2 Matrixmatrix.
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Step 5 – Likewise do it for a2 and a3.
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Step 6 – Sum all of them, do not forget the minus signs before
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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.
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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.
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