Singular Matrix Example, Applications and Types

A matrix is an ordered array of numbers and elements that can be arranged in different ways. Each number or variable in the matrix is called an element. The size of a matrix is determined by the number of rows and columns in it. It is also known as the dimension of the matrix. Any matrix whose determinant is zero is called a singular matrix. A singular matrix means that it will not have any multiplicative inverse, i.e., they are non-invertible. In this case, no inverse exists, which means A^-1 is not defined. A singular matrix crushes at least one dimension in such a way that information will completely disappear, and it is not possible to retrieve information. 

Exams like IIT JAM and CUET may not ask direct questions on examples. However, those who can identify different matrices can solve even complex questions. Let us take a look at the types of singular matrices:

Matrices chapter is one of the important topics for most entrance exams like JEE Main and NEET, among others. It is therefore, important to understand the applications of these matrices:

• Linear Algebra and Systems of Equations: Singular matrix helps to understand the nature of solution in linear algebra since it can indicate that a linear equation system has either no solution or it has infinitely many solutions.
• Computer Graphics: A singular matrix can project a 3D object onto a 2D plane, which is required for rendering graphics.
• Economics and Input-Output Models: In economic modelling, a singular matrix is used for input-output models where some industries cannot produce output independently.
• Multicollinearity: In regression analysis, singular matrix indicates multicollinearity among independent variables. This indicates that some of the variables are linearly dependent on others. This impacts the estimation of regression coefficients.
• Degenerate Systems: In physics, singular matrices can describe those systems that are in degenerate state. This is especially helpful where multiple states correspond to the same energy level.

Let us understand the difference between singular and non-singular matrix:

Let us look at some illustrative examples for students who are preparing for entrance exams like IISER and GATE:

1: For the singular matrix A, (A′) ^–1 = (A^–1)′. Is state true or false?

Solution: False we know that the singular matrix is non-invertible.

2: If A and B are two skew-symmetric matrices of the same order, AB is a symmetric matrix.

Solution: AB = BA

3: What is the determinant of any given singular matrix?

Solution: The determinant of any given singular matrix is always 0.