Application of Matrices in Science, Commerce and Social Science Fields

Matrix applications are widely used in mathematics as well as other subjects. It aids in the solution of linear equations. Matrices are incredibly valuable items that can be found in a variety of settings. The usage of matrices in mathematics can be found in a wide range of scientific and mathematical subjects. Almost every element of our life is influenced by engineering mathematics. In this post, we'll go over what a matrix is, how to use matrices, and how to solve problems using matrices.

They are used in computer graphics to project a three-dimensional image onto a two-dimensional screen. Stochastic matrices are used to explain sets of probabilities in probability theory and statistics; for example, they are utilised in the page rank algorithm that ranks the sites in a Google search.

What are Matrices?

• A matrix is defined as a rectangular array of numbers or symbols which are generally arranged in rows and columns.

• The order of the matrix can be defined as the number of rows and columns.

• The entries are the numbers in the matrix known as an element.

• The plural of a matrix is matrices.

• The size of a matrix is denoted as ‘n by m’ matrix and is written as m×n, where n = number of rows and m = number of columns.

Types of Matrix

There are different types of matrices. Here they are:

1) Row matrix

2) Column matrix

3) Null matrix

4) Square matrix

5) Diagonal matrix

6) Upper triangular matrix

7) Lower triangular matrix

8) Symmetric matrix

9) Anti-symmetric matrix

\[A = \begin{bmatrix}                  1 & 2 &3 \\  7& 8 & 9\end{bmatrix}, B = \begin{bmatrix}5 & 6 & 7 \\ 3 & 4 & 5\end{bmatrix}\]

\[A+B = \begin{bmatrix} 1+5 & 2+6 & 3+7\\ 7+3 & 8+4 & 9+4 \end{bmatrix}\]

\[A\div B = \begin{bmatrix} 6 & 8 & 10\\ 10 & 12 & 14 \end{bmatrix} \]

Applications of Matrices

Matrices have many applications in diverse fields of science, commerce and social science. Matrices are used in:

(i) Computer Graphics 

(ii) Optics 

(iii) Cryptography 

(iv) Economics

(v) Chemistry 

(vi) Geology 

(vii) Robotics and animation 

(viii) Wireless communication and signal processing 

(ix) Finance ices

(x) Mathematics

Use of Matrices in Computer Graphics

Earlier, architecture, cartoons, and automation were done by hand drawings but nowadays they are done by using computer graphics. Square matrices very easily represent the linear transformation of objects. They are used to project three-dimensional images into two-dimensional planes in the field of graphics. In graphics, a digital image is treated as a matrix to start with. The rows and columns of the matrix correspond to rows and columns of pixels and the numerical entries correspond to the pixels’ colour values.

Using matrices to manipulate a point is a common mathematical approach in video game graphics. Matrices are also used to express graphs. Every graph can be represented as a matrix, each column and each row of a matrix is a node and the value of their intersection is the strength of the connection between them. Matrix operations such as translation, rotation and sealing are used in graphics. 

Use of Matrices in Cryptography

Cryptography is the technique to encrypt data so that only the relevant person can get the data and relate information. In earlier days, video signals were not used to encrypt. Anyone with a satellite dish was able to watch videos, which resulted in the loss for satellite owners, so they started encrypting the video signals so that only those who have video cyphers can unencrypt the signals. 

This encryption is done by using an invertible key that is not invertible then the encrypted signals cannot be unencrypted and they cannot get back to their original form. This process is done using matrices. Digital audio or video signal is firstly taken as a sequence of numbers representing the variation over time of air pressure of an acoustic audio signal. The filtering techniques are used which depend on matrix multiplication. 

Use of Matrices in Wireless Communication

Matrices are used to model the wireless signals and to optimise them. For detection, extractions and processing of the information embedded in signal matrices are used. Matrices play a key role in signal estimation and detection problems. They are used in sensor array signal processing and the design of adaptive filters. Matrices help in processing and representing digital images. 

We know that wireless communication is an important part of the telecommunication industry. Sensor array signal processing focuses on signal enumeration and source location applications and presents huge importance in many domains such as radar signals and underwater surveillance. The main problem in sensor array signal processing is to detect and locate the radiating sources given the temporal and spatial information collected from the sensors.

Use of Matrices in Science

Matrices are used in the science of optics to account for reflection and for refraction. Matrices are also useful in electrical circuits and quantum mechanics and resistor conversion of electrical energy. Matrices are used to solve AC network equations in electric circuits.

Application of Matrices in Mathematics

The application of matrices in mathematics has  an extended history of application in solving linear equations. Matrices are incredibly useful things that happen in many various applied areas. The application of matrices in mathematics applies to many branches of science, also as different mathematical disciplines. Engineering Mathematics is applied in our daily life.

Use of Matrices for Collinear Point

Matrices can be used to check whether any three given points are collinear or not. Three points suppose A(a,b), B(c,d), C(e,f) are collinear if they do not form a triangle, that is the area of the triangle should be equal to zero. 

Use of Matrices in Social Science

One-dimensional information, such as a family's total monthly cost, can be conveyed using real numbers. However, if two families' monthly spending on three items—food, entertainment, and health (indexed by 1, 2, 3)—are to be recorded, a rectangular array of real values, or a matrix, must be used. 

A matrix (A) is a rectangular array of numbers, parameters, or variables that can be used to solve problems. The elements of the matrix are the array's members, and they're commonly surrounded in brackets, parentheses, or double vertical lines.

Uses of Matrices in Commerce

Matrix Cramer's Rule and determinants are useful tools for resolving various problems in business and economics involving profit maximisation and loss minimization. Variance and covariance are calculated using matrices. With the use of a matrix determinant, Matrix Cramer's Rule is utilised to find solutions to linear equations. The IS-LM model's market equilibrium is solved with determinants and Matrix Cramer's Rule.