Matrices Introduction:The meaning of a matrix is a mix of ranges structured into a hard and fast number of rows and columns. Generally, the numbers are real numbers but it may involve complex numbers too. In the example below you can see (3×3) three columns and three rows. The numbers in the matrix are called elements of the matrix. Vertex matrices are used the business realm. Vatrix matrices are graphs showing joined concepts or data. It’s exceptionally helpful for owners who have businesses, they’re able to visualize the company as a whole, including relationships among different business apartment and the total efficiency of the workforce. It can also upgrade customer service by decreasing the procedures to greet a customer and deliver a product or service helps a business to increase its cash flow and deliver a better purchase experience. The owner also can smooth the path to completion for his staff by removing any unneeded vertex or two from the larger matrix. The outcome of this is having a more efficient work force and lightens the drag on the company’s revenue stream. Matrices apply to a variety of branches of science, it’s used also in every computer generated image that has a reflection, or distortion effects such as light passing through rippling water. Backround:Matrices started in the second century BC and changed the way people use math till this day. It was originally started by the babylonians but then the chinese used it to make things much easier and even improved the system. Matrices was mostly used to figure out the size of the fields and later on used to measure how much corn is in the bundle. But now a days it is used to get linear equations. On 1545 Cardan made a new rule that was formed to solve two linear equations. But his theory was very close to Cramer’s theory. This was a major transformation for the theory because in 1730 Maclaurin changed how the system works by saying that Cramer’s theory could work and that 2×2 and 3×3 and 4×4 would be more efficient. Gauss introduced the word determinant in 1801. The determinant was used to get the quadratic form. In the year 1826, Cauchy found the eigenvalues and gave results on diagonalisation of a matrix in the context of converting a form to the sum of squares. The first person to use the term matrix was Sylvester in 1850. Then Cayley in 1858 published Memoir on the theory of matrices for containing the first abstract definition of a matrix.On the year 1870 the Jordan canonical form came to get a prime.But in 1878 a very important report was wrote by Frobenius that made a huge impact.In the the report he proved that the matrix satisfies the characteristic equation.In 1876 Forbinus started using the old way even though it was not proved and made in better.He used Cayley’s 2×2 way of solving. So no matter what new ways of solving matrices were made they went back to the old way and improved it to a way that people still use it today.Discussion: The Matrix is very useful and essential in a lot of places such as math, science and the business world. Matrix plays a big role in calculating electrical properties in a circuit and also in voltage and resistance. It has advantages and disadvantages. Some advantages in the business world is that it joins together teamwork, accountability, and redundancy. In teamwork you put all the employees together and this forms a group where they work together and share ideas and criticize each other’s work. When working in redundancy the workers will be able to specialise in certain parts and get the best results. Also, there are a lot of disadvantages like not knowing who is the boss. It can cause credible problems because the group members and workers do not know who to take orders from or who is the boss, which may often cause workers to slack off. Another disadvantage is wages or salaries once all the workers receive the same amount they do not find the urge to work harder and try to get a promotion because all are equal like communists.As you can see that the advantages and disadvantages relate to math because this shows how employees or workers work in the same group like an equation and all receive the same value or amount. So all and all you can say that there are ups and downs in using the matrix system and how it can be used in real life or in companies. Although, in the math world there are some situations where using matrices for linear transformation to render images. Application: One example where matrices can be used and applied is in graphic software. Graphic software uses matrix mathematics to process linear transformation to render images. A square matrix contain the same numbers of rows and columns can illustrate a linear transformation of a geometric object. For example, the cartesian X-Y plane, the matrix indicates an object in the vertical Y axis. In a video game, this would render the upside-down mirror image of a castle reflected in a lake. It would be intricate if the surfaces are curved and reflected, such as shiny silver goblet, it’s harder to stretch or shrink the reflection. The matrix arithmetic helps us calculate the electric properties of a circuit with amperage, voltage resistance, etc. A matrix organizes a group of numbers, or variable with specific rules of arithmetic. It’s illustrated like this this 2×3 matrix has two rows and three columns. This is an example of a square matrix with variables and it’s a square matrix because the rows and columns are equal. We can only add matrices of the same dimensions, because we add the corresponding element. Matrix multiplication is different. For example multiplying MP=R. M is an mXn matrix; P is nXp; and the result R will have dimension mXp. Also the number of columns of the left-hand matrix, M, must equal the number of rows of the right hand matrix, P. Cryptography is an application of matrices. Cryptography is responsible with keeping communications private. It is very vital for business companies to use cryptography. Three reasons include, firstly it follows standards for encryption and key management, secondly it’s easy to establish and utilize, and thirdly it delivers strong security against data thieves. Cryptography includes encryption and decryption. Encryption is modifying the data into an illegible form, their use is to obscure information from anyone who is able to read it. Decryption conflicts with encryption, whereas decryption it transforms the encrypted data back to its form. Application of matrix to cryptography first a one type of code, which is strenuous to break, makes use of a large matrix to encode a message. Then the receiver of the message decodes it using the inverse of the matrix. So the first matrix which is used by the sender is called the encoding matrix and its inverse is called the decoding matrix, which is used by the receive