Singular Matrix

A matrix is an ordered arrangement of rectangular arrays of function or numbers, that are written in between the square brackets. Each row and column include the values or the expressions that are called elements or entries. The total number of rows by the number of columns describes the size or dimension of a matrix. The matrix representation is as shown below.

singular Matrix

The matrix shown above has m-rows (horizontal rows) and n-columns ( vertical column). The order of the matrix is given as m \(\times\) n.

Types of matrices

We have different types of matrices, such as a row matrix, column matrix, identity matrix, square matrix, rectangular matrix. One of the types is a singular Matrix.

What is Singular Matrix?

A square matrix (m = n) that is not invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is 0.

If we assume that,

A and B are two matrices of the order, n x n satisfying the following condition:

AB = I = BA

Where I denote the identity matrix whose order is n.

Then, matrix B is called as the inverse of matrix A.

Therefore; A is known as a non-singular matrix.

The matrix which does not satisfy the above condition is called as a singular matrix .i.e. a matrix whose inverse does not exist.

Determinant

Every square matrix has a determinant. The determinant is a mathematical concept that has a vital role in the solution as well as analysis of linear equations.

Suppose A is a matrix as given below

\(\large A = \begin{bmatrix} a & b & c\\ d & e & f\\ g & h & i \end{bmatrix}\)

The determinant of the matrix A is denoted by |A|, such that

\(\large \begin{vmatrix} A \end{vmatrix} = \begin{vmatrix} a & b & c\\ d & e & f\\ g & h & i \end{vmatrix}\)

It can be calculated by:

The determinant can be calculated as

\(\large \begin{vmatrix} A \end{vmatrix} = a(ei – fh) – b(di – gf) + c (dh – eg\)

For a Singular matrix, the determinant value has to be equal to 0, i.e. |A| = 0.

Thus, a(ei – fh) – b(di – fg) + c(dh – eg) = 0

Example: Determine whether the given matrix is a Singular matrix or not

\(\mathbf{\begin{bmatrix} 2 & 4 & 6\\ 2 & 0 & 2 \\ 6 & 8 & 14 \end{bmatrix}}\).

Solution: Given \( \begin{bmatrix} 2 & 4 & 6\\ 2 & 0 & 2 \\ 6 & 8 & 14 \end{bmatrix}\)

The Determinant is given by-

\( 2(0 – 16) – 4 (28 – 12) + 6 (16 – 0) = -6(16) + 6 (16) = 0\)

As the determinant is equal to 0, hence it is a Singular Matrix.

We already know that for a Singular matrix, the inverse of a matrix does not exist. Let us learn why the inverse does not exist.

The inverse of a matrix ‘A’ is given as- \(\mathbf{A’ = \frac{adjoint (A)}{\begin{vmatrix} A \end{vmatrix}}}\)

for a singular matrix \(\begin{vmatrix} A \end{vmatrix} = 0\),,

the denominator term needs to be 0 for a singular matrix, that is not-defined.

Therefore, the inverse of a Singular matrix does not exist.

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Practise This Question

Which term of the AP: 3, 15, 27, 39, ... will be 132 more than its 54th term?