Matrix is defined as the array of numbers in the form of a rectangle ( the numbers can be real/complex), symbols or expressions arranged in rows and columns.
It is denoted by A.
A is a rectangular matrix with m rows and n columns. The numbers of this array are called its elements.
A matrix can be denoted by using the bracket ( ) or [ ].
The term Matrix was designed in the century by the English Mathematician James Sylvester and his friend Arthur Cayley, a mathematician who developed the algebraic aspect of matrices in the 1850s.
The invention of the matrix was due to the motivation behind the attempts to develop the right algebraic language to study the determinants.
The plural form of a matrix is called matrices.
A matrix A with m rows and n columns is called a matrix of order (m, n) or m × n (read as m by n).
Consider the matrix A=
It is a matrix of order 3 × 3. Here the 3 occurs in the third row and second column. Element 9 appears in the second row and third column. In notations, we may write it as =3 and =9.
Types of Matrices
The below listed are some of the widely used matrices:
- Row Matrix
- Column Matrix
- Zero Matrix or Null Matrix
- Square Matrix and Rectangle Matrix
- Scalar Matrix
- Unit Matrix
- Upper triangle Matrix
- Lower triangle Matrix
- Sub Matrix
- Equal Matrices
Algebra of Matrices
- Addition and Subtraction of Matrices
Let A and B be the two matrices of the same order. Then the addition of A and B will be denoted by A B. This matrix is obtained by adding the corresponding entries of A. Similarly, in subtracting two matrices, their corresponding elements are subtracted.
Remark: We can add two matrices of the same order. If they are of the same order, they are more comfortable for an additional purpose. Also, the order of the matrices is the same as that of the original two matrices.
If A, B, C are matrices of the same order, then the following are the properties:
- A + B = B + A (Commutative Law)
- (A + B) + C = A + (B + C) (Associative Law)
- K(A + B) = K.A + K.B, where m is a constant
- Negative of a Matrix
If A is any matrix, then the negative of the A is denoted by as follows:
-A =
- Multiplication of Matrices
The product AB of two matrices, A and B, will be defined only if the numbers of the columns in Matrix A is equal to the number of rows in Matrix B.
Some properties of multiplication of matrices are as follows:
- Multiplication of matrix is generally not commutative, which means that AB ≠ BA.
- Multiplication of matrix is associative, i.e. (AB)C = A(BC), where both sides are defined.
- If A, B, and C are three matrices making AB = AC, then generally B ≠ C.
- The product of two non zero matrices will be non zero matrices.
Real-Life Applications of Matrices
Some of the Real Life Applications of Matrices are:
Matrices have many uses in a wide variety of fields, whether it be in science or commerce. Some of such major applications of matrices in real life have been discussed below-
- Robotics and Automation
Matrices form the base elements for the movements of robots. The movements and inputs for controlling the robots are programmed using rows and columns of matrices.
- Physics
The study of electrical circuits, optics and quantum mechanisms requires the application of matrices. They are also used in resistor conversion of electrical energy into other helpful energy, calculating battery power outputs and solving Kirchoff’s laws of voltage and current.
- Computer graphics and coding
In computer-based applications, matrices play a vital role in the projection of 3D images into 2D. They enable the conversion of geographic data into different coordinate systems. Eigen Vector Solvers is used to rank the web pages in the google search.
- Cryptography
Programmers use them to code, encrypt and decrypt messages. A message is made as a sequence of numbers for communication in binary form following a code theory for solving, solved using a key matrix.
This key matrix must be kept secret between the sender and the receiver, or else unauthorised people would easily decode the messages.
- Engineering
The numbers in matrices can represent data and mathematical equations as well. In many time-sensitive engineering applications, matrices provide quick and better approximations of many complicated calculations.
- Chemistry
Matrices are used to determine the coefficients of a balanced chemical reaction through a system of linear equations.
- Geology
In geology, matrices are used in taking seismic surveys. A seismic survey is a method used to gather information about the location and characteristics of geological structures beneath the earth’s surface.
- Representing world data
Matrices represent various data such as the population of people, infant mortality rate, etc.
- Business
To determine which combinations of ingredients are to be used in a product to produce the most profitable one, matrices are used.
They also help in the decision-making process in a business. It helps a company in visualising the interconnected relationships between its various departments.
- Finance
Matrix is used in financial risk management and to determine the outcomes or payoff of an enterprise or any investment. According to the calculated risk, the investor can invest in a project or venture within their risk-bearing capacity.
- Economics
Economists use matrices to calculate the Gross Domestic Product (GDP) of a particular nation produced during a given period. By calculating GDP, the market value of all the final economic goods and services produced within the domestic territory of a country can be determined.
The government of such a country can work towards the betterment of the country, considering the GDP.
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- Banking sector
Because of the use of matrices, the banks work with the transmission of sensitive and private data. The internet works with the help of matrices, and in this modern era, banks are very much dependent on the internet for its various functions.
- Communication through Wireless medium
Matrices are used in detecting, extracting and processing information that’s embedded in the wireless signals. They also help in detecting any problems that occur in such signals.
Conclusion
The article makes it immensely clear that matrices are of great importance in everyday life. We have been so accustomed to their use that we often seem to ignore them.
