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Singular Matrix

Before knowing about a singular matrix, let us first know about a matrix itself.

So, a matrix is a rectangular array or table of symbols, numbers, or expressions arranged in rows and columns. This arrangement of numbers is enclosed between square brackets. The general representation of the order of the matrix is m x n, where m is the number of rows and n is the number of columns.

A representation of a matrix - Singular Matrix
A representation of a matrix (Source)

What is a Singular matrix?

Coming to the definition of a singular matrix, it is basically a non-invertible square matrix i.e the determinant of this square matrix is 0.

Now, a square matrix is a matrix that has an equal number of rows and columns, i.e., m = n. 

An invertible matrix is a square matrix that satisfies the condition:

Product of the matrix and its inverse = Identity matrix

If A is the actual matrix, B is its inverse and I is the identity matrix then,

A x B = I

Determinant of a matrix, A is denoted by |A|.

The determinant of a matrix is calculated as follows,

\(
\det \begin{bmatrix}
a & b\\
c & d
\end{bmatrix} = ad – bc\\
\)

For singular matrices, |A| = 0

Properties

  • The determinant is 0.
  • It is defined only for non-invertible square matrices.
  • There is no multiplicative inverse for this matrix.

Solved Examples

Question 1. Check if the following matrix is singular or not.

\(
\begin{bmatrix}
1 & 2\\
2 & 4
\end{bmatrix}
\)

Solution. Let us call the above matrix, A.

.Let us check its determinant

⇒ det(A) = 4*1 – 2*2 = 0

So, as the determinant comes out to be zero, by definition it is singular.

FAQs

What is a singular matrix?

A singular matrix is a matrix that is a square matrix, is non-invertible, and has a determinant 0.

How to know if a matrix is singular?

It should be a square matrix and should have determinant 0 to be singulat

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