Singular Matrix: Properties, Importance and Determinant

Singular matrices are non-invertible square matrices. A square matrix is considered a singular matrix only if its determinant is zero. It is a matrix that does not have an inverse matrix. In this article, we will understand more about Matrices, types of Matrices, Singular Matrix and Determinants. 

Keyterms: Square Matrices, Matrices, Determinants, Non-Singular Matrix, Inverse Matrix, Row, Column Matrix, Singular Matrix

Read Also: Matrix Addition

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Matrix

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A matrix is an ordered collection of rectangular arrays of functions or integers placed between square brackets. Elements or entries are the values or expressions that appear in each row and column. The size or dimension of a matrix is defined by the total number of rows divided by the total number of columns. The matrices are represented as follows.

There are m-rows (horizontal rows) and n-columns in the matrix illustrated above ( vertical column). The matrix's order is written as m x n.

The video below explains this:

Matrices Detailed Video Explanation:

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Types of Matrices 

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In mathematics, there are various types of matrices, such as:

• Singular Matrix
• Column Matrix
• Row matrix
• Identity Matrix
• Rectangular matrix
• Square matrix

Also Read: Maxima and Minima

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What is a Singular Matrix?

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Singular or degenerate matrices are non-invertible square matrices. A square matrix is singular only if its determinant is zero.

Assume P and Q are two (2) matrices of the order an x a that satisfies the following condition:

PQ = I = QP

The ‘Identity matrix,' whose order is ‘a,' is represented by ‘I.'

Matrix Q is thus known as the inverse of matrix P.

As a result, P is referred to as a non-singular matrix.

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What is a Determinant?

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A determinant exists for every square matrix. It's a mathematical idea that's crucial for solving and analyzing linear equations.

Assume that ‘M' is a matrix of the form:

M = ( a b c d e f g h i)

The determinant of the matrix M is denoted by the symbol |M|, as follows:

M = |a b c d e f g h i|

The determinant can be assessed as follows:

|M| = a (ei – fh) – b (di – gh) + c (dh – eg)

The value of the determinant must equal 0 to determine a Singular matrix, i.e. |M| = 0.

So,

a (ei – fh) – b (di – gh) + c (dh – eg) = 0

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Properties of Singular Matrix 

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• A matrix is considered to be a singular matrix if its determinant equals 0. For example, a matrix x with zero members in the first column.
• The determinant in this example is 0, according to the principles and properties of determinants. As a result, matrix x is unquestionably a singular matrix.
• In nature, a singular matrix is non-convertible.
• The inverse of a singular matrix does not exist.

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Importance of a Singular Matrix

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• Singular matrices serve as a dividing line between matrices with positive determinants and those with negative determinants. The determinant's sign has ramifications in a variety of domains.
• Multiplication by matrices with a positive determinant, for example, results in the preservation of the orientation. Multiplication by matrices having a negative determinant, on the other hand, results in the reversal of orientation.
• The determinant sign relative to the trace clearly influences the qualitative behavior of non-linear ordinary differential solutions. The determinants of singular matrices are neither positive nor negative. This distinguishes them from the rest.
• Singular matrices would occur with vanishing probability in actual life. In addition, matrices with determinants near 0 are very common in real life.
• They behave in a way that is surprisingly similar to singular matrices. As a result, learning about singular matrices is crucial.

Also Read: Integrals

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Difference between Singular Matrix and Non-Singular Matrix

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• Square matrices contain unique characteristics that distinguish them from other matrices. The number of rows and columns in a square matrix is the same. Singular matrices are one-of-a-kind and can't be multiplied by another matrix to produce the identity matrix. Non-singular matrices are invertible, which allows them to be used in various linear algebra procedures like singular value decompositions. Identifying if you're working with a singular or non-singular matrix is the initial step in many linear algebra tasks.
• Determine the matrix's determinant. The matrix is singular if and only if it has a determinant of zero. The determinants of non-singular matrices are non-zero.
• Determine the matrix's inverse. If a matrix has an inverse, multiplying the matrix by its inverse yields the identity matrix. The identity matrix is a square matrix with the same dimensions as the original matrix and zeroes on the diagonal. The matrix is non singular if an inverse can be found for it.
• To prove that the matrix is non-singular, make sure it fits all of the other conditions for the invertible matrix theorem. The determinant of a "n by n" square matrix should be non-zero, the matrix's rank should equal "n," the matrix should have linearly independent columns, and the matrix's transpose should be invertible.

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Things to Remember

• The determinant of a matrix is said to be singular if and only if it is equal to zero. 
• A singular matrix is one that has no inverse and thus no multiplicative inverse.
• A matrix's size is known as a 'n by m' matrix and is written as nm, where n is the number of rows and m is the number of columns.
• For example, we have a 4 × 6 matrix because the number of rows equals 4 and the number of columns equals 6.

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