Singular Matrix

A matrix is an array containing elements which are arranged in rows and columns. Based on the arrangements of elements in rows and columns, matrices are classified into different types like 2 x 2 matrices , 3 x 3 matrices, 2 x3 matrices etc. A 2x 3 matrix has two rows and three columns.

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A matrix is a rectangular arrangement of numbers
example,

Matrix(A)

An alternative notation uses large parentheses instead of box brackets:

Matrix(B)

Rows and columns are nothing but the horizontal and vertical lines in a matrix. The numbers in the matrix are called its elements. To specify a matrix's size, a matrix with m rows and n columns is called m-by-n matrix or m × n matrix, while m and n are called its rows and columns.
Types of Matrices - Singular Matrix

A matrix may be classified by types.

1. Row matrix

2. Column matrix

3. Zero matrix

4. Square matrix

5. Diagonal matrix

6. Unit matrix:

A square matrix is singular if and only if determinant of matrix is zero.

. A matrix is singular iff its determinant  is 0

If the determinant of a  square matrix is 0 then the matrix has no inverse , It is called a singular matrix

A square matrix is singular if and only if its determinant is zero

A matrix is singular iff its determinant  is 0

If the determinant of a  square matrix is 0 then the matrix has no inverse , It is called a singular matrix

Square matrix:

A square matrix is a matrix which has the same number of rows and columns. An n-by-n matrix is called as a square matrix of order n. We can add any two square matrices of the same order

An example of a square matrix is :

Example of square matrix

This matrix has 3 rows and 3 columns: m=n=3.

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Determinant:

The determinant det(A) or |A| of a square matrix A  is a number encoding certain properties of the matrix. A matrix is invertible if and only if determinant of that matrix is nonzero.

The determinant of 2-by-2 matrices is given by

Formula for determinant
For 3x3 matrices the formula is more complicated:

Determinant of matrix

Properties of Determinants - Singular Matrix

There are three rules that all determinants follow. These are:

1. The determinant of an identity matrix is 1

2. If two rows or two columns of the matrix are exchanged, then the result is determinant multiplied by -1.

3. If all the numbers in one row or column are multiplied by a number n, then the determinant is multiplied by n. Also, if a matrix M has a column v that is the sum of two column matrices v1 and v2, then the determinant of M is the sum of the determinants of M with v1 in place of v and M with v2 in place of v. These two conditions are called as multi-linearity.
Example:

Solution:

Determinant =  (ad-bc)

= (3 × 2) – (6 × 1) = 0

The given matrix does not have an inverse. It is singular matrix.