Application of Matrices

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 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.

  • Example:


\[\begin{bmatrix} 6  & 4  & 24\\ 1  & -9 & 8 \end {bmatrix}\]

The matrix given above has 2 rows and three columns.

Types of Matrix

What are The Different 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

Important Operations on Matrices:

  1. Addition of Matrices:

Let us suppose that we have two matrices namely A and B.

Both the matrices A and B have the same number of rows and columns (that is the number of rows is 2 and the number of columns is 3), so they can be added. In simpler words, you can easily add a 2 x 3 matrix with a 2 x 3 matrix or a 2 x 2 matrix with a 2 x 2 matrix. However, remember you cannot add a 3 x 2 matrix with a 2 x 3 matrix or a 2 x 2 matrix with a 3 x 3 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 = \[\begin{bmatrix} 1 + 5  & 2 + 6  & 3 + 7\\ 7 + 3  & 8 + 4 & 9 + 5 \end {bmatrix}\]

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

Note:  Keep in mind that the order in which matrices are added is not important; thus, we can say that  A + B is equal to B + A.

  1. Multiplication of Matrices :

We will consider a simple 2 × 2 matrix multiplication A= \[\begin{bmatrix} 3  & 7 \\ 4  & 9 \end {bmatrix}\] and another matrix B= \[\begin{bmatrix} 6  & 2 \\ 5  & 8 \end {bmatrix}\]

 

Now we can calculate each of the elements of product matrix AB as follows:

  • Product of AB11 = 3 × 6 + 7 ×5 = 53

  • Product  of AB12 = 3 × 2 + 7 × 8 = 62

  • Product of AB21 = 4 × 6 + 9 × 5 = 69

  • Product  of AB22 = 4 × 2 + 9 × 8 = 80

Therefore matrix AB is equal to,

 

AB = \[\begin{bmatrix} 53  & 62 \\ 69  & 80 \end {bmatrix}\]

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, automation were done by hand drawings but nowadays they are done by using computer graphics. Square matrices very easily represent linear transformation of objects. They are used to project three dimensional images into two dimensional planes in the field of graphics. In Graphics, 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’ color 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. For transformation of a point we use the equation

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 satellite dish was able to watch videos which results in the loss for satellite owners, so they started encrypting the video signals so that only those who have video ciphers 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. A 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 depends on matrix multiplication. 

Use of Matrices in Wireless Communication

Matrices are used to model the wireless signals and to optimize them. For detection, extractions and processing of the information embedded in signals matrices are used. Matrices play a key role in signal estimation and detection problems. They are used in sensor array signal processing and design of adaptive filters. Matrices help in processing and representing digital images. We know that wireless and communication is an important part of the telecommunication industry. Sensor array signal processing focuses on signal enumeration and source location applications and presents a huge importance in many domains such as radar signals and underwater surveillance. 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 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

Application of matrices in mathematics have an extended history of application in solving linear equations. Matrices are incredibly useful things that happen in many various applied areas. 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 in Finding Area of Triangle

We can use matrices to find the area of any triangle where the vertices of the triangle have been given.

Let’s suppose that we have a triangle ABC with vertices A(a,b) , B(c,d) , C(e,f)

Now the area of the triangle ABC ,


Area of triangle ABC can be given by the determinant= \[\frac{1}{2} \begin{bmatrix} a  & b & 1\\ c  & d & 1 \\ e & f  & 1\end {bmatrix}\]


Use of Matrices for Collinear Point

Matrices can be used to check where 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.

The points A,B,C are collinear if \[\begin{bmatrix} a  & b & 1\\ c  & d & 1 \\ e & f  & 1\end {bmatrix}\] vanishes.


FAQ (Frequently Asked Questions)

1. What are the applications of matrices?

They are used for plotting graphs, statistics and also to do scientific studies and research in almost different fields. Matrices can also be used to represent real world data like the population of people, infant mortality rate, etc. They are the best representation methods for plotting surveys.

2. What is the application of matrices in engineering?

Application of matrices in Engineering

Transformation matrices are commonly used in computer graphics and image processing. Matrices are used in computer generated images that have a reflection and distortion effect such as high passing through ripping water. This is how Application of matrices in engineering is used.


3. What is the application of matrices in business and economics?

The idea of application of matrices in business is that you have multiple inputs and you have multiple outputs, and you are trying to predictively model changes in dependent variables, by examining the impact of numerous independent variables that do provide explanation. The coefficient factors of the variables used lend themselves to a matrix form. This is how application of matrices in business is used.