Introduction to Matrices
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.
FAQs on Application of Matrices in Science, Commerce and Social Science Fields
1. What are the major applications of matrices in diverse fields like science, commerce, and social sciences?
Matrices are mathematical structures used to organise data in rows and columns, making them incredibly versatile. Their primary applications include:
- Science & Engineering: Used in optics for refraction, in electrical circuits to solve network equations, and in quantum mechanics. They are also fundamental to wireless communication and signal processing.
- Commerce & Economics: Used to solve systems of linear equations for profit maximisation, in business to model inputs and outputs, and to calculate variance and covariance in statistics.
- Computer Science: Essential for computer graphics, cryptography, and representing data in graphs and networks.
- Social Sciences: Used in sociology to study social networks and in economics to model population shifts or economic flows.
2. How are matrices specifically used in computer graphics and animation?
In computer graphics, transformation matrices are used to manipulate 2D and 3D objects on a screen. Every point of an object can be represented as a matrix, and operations like translation (moving), rotation (turning), and scaling (resizing) are performed by multiplying the object's matrix with a specific transformation matrix. Digital images themselves are treated as matrices where each element corresponds to a pixel's colour value.
3. What is the main application of matrices and determinants according to the CBSE Class 12 Maths syllabus for 2025-26?
As per the CBSE 2025-26 syllabus, the primary application of matrices and determinants is to analyse and solve systems of linear equations. This includes:
- Checking the consistency of a system of equations (whether it has a unique solution, infinite solutions, or no solution).
- Solving a system of linear equations in two or three variables using the inverse of a matrix.
- Applying these methods to solve real-world problems that can be formulated as a system of linear equations.
4. Explain the role of matrices in the field of economics.
In economics, matrices are used to model complex economic systems. For instance, they help in solving problems related to market equilibrium using the IS-LM model. Business strategists use them for linear programming to determine the best allocation of resources for profit maximisation and loss minimisation. Furthermore, tools like Cramer's Rule, which relies on matrix determinants, are used to solve for variables in economic models.
5. How do matrices help secure information in cryptography?
Cryptography uses matrices to encrypt data. A message is first converted into a matrix of numbers. This message matrix is then multiplied by a special invertible matrix known as the 'key'. The resulting matrix is the encrypted message. To decrypt it, the receiver multiplies the encrypted message by the inverse of the key matrix. Without the exact key, it is extremely difficult to decode the message, thus ensuring data security.
6. Why are matrices considered a superior tool for representing systems of linear equations?
Matrices are superior because they provide a compact and systematic way to manage multiple variables and equations at once. Instead of writing out long equations, a system can be represented as a single matrix equation (AX = B). This simplifies complex calculations into structured operations. Properties of the matrix, such as its determinant or inverse, directly reveal crucial information about the system, like whether a unique solution exists, making the entire process highly efficient and less prone to error.
7. What fundamental property makes a problem suitable to be modelled by a matrix?
A problem is suitable for a matrix model if it involves a system of linear relationships between multiple inputs and outputs or data points that can be organised in a grid. This applies to various scenarios, such as a set of simultaneous linear equations, the coordinates of a geometric shape, connections in a network, or the pixel data of an image. The key is the structured, grid-like arrangement of data or variables that allows for systematic mathematical operations.
8. How can matrices be used in social sciences to study relationships?
In social sciences, particularly in sociology and anthropology, matrices are used in social network analysis. An 'adjacency matrix' can represent a network of people, where each row and column corresponds to a person. The value at the intersection (e.g., a 1 or 0) indicates the presence or absence of a relationship (like friendship or communication). By performing matrix operations, researchers can analyse the structure of the network, identify influential individuals, and study how information flows through the group.
