Matrices economy, and even atomic physics. Because of

Matrices were introduced in 1850 by a famous English mathematician named James Joseph Sylvester. He started by writing down an arrangement of terms and extracted rows and columns from it. Sylvester formed it into a rectangular array, and he called it a matrix. Soon after, mathematicians recognized that rectangular arrays are convenient devices for extending the common notions of number. Arthur Cayley also made great contributions to the subject of matrices in the 19th century. Arthur Cayley applied matrices to systems of linear equations and realized that certain sets of matrices can form algebraic systems. He defined operations such as addition, subtraction, multiplication and division as transformations of matrices. The theory of matrices is a part of algebra, but it became increasingly clear that matrices have had a purpose that extends beyond the field of algebra and into other parts of mathematics. It was found that they were necessary for expressing many ideas of applied mathematics. In 1925, Heisenberg introduced matrices into quantum mechanics.I chose the topic of Real Life Applications of Matrices because of the many different and interesting ways that matrices can be applied to everyday things, and how useful they are. They are extremely useful in many branches of engineering, science, economy, and even atomic physics. Because of how relevant these different examples are, the topic of matrices interested me very much. Matrices are one of the most interesting and important parts of mathematics.A matrix is a rectangular array of numbers, which is arranged in rows and columns. The use of matrices has now extended beyond mathematics and the physical sciences to business, psychology, and social and political sciences, and more. Matrices are also used in things like game theory, graph theory, transformations of a coordinate plane, large systems of equations and more. Matrices are read by row number, then column number (row, column). In matrix A, seen below, it is a 3 by 3 matrix, since it is composed of 3 rows and 3 columns.A matrix entry is an individual number in a matrix. A matrix element is one of the matrix entries. Matrix entries are identified by specifying the row and column in which they are in. Since this matrix name is A, the symbol used to denote the entry is Ai,j. The i denotes the row number and the j denotes the column number. For example, if you were to express the location of the value of 0 in matrix A, the element would be named A2,3. Two matrices are only considered equal if they are of the same size, and all entries are corresponding of one matrix to the other. Under the definition of equality in matrices, equation A can also be written as the equation B.Matrices are able to be added and multiplied. To add two matrices together that are the same size, for example, matrices A and B, it is defined by the formula A + Bi,j =  Ai,j + Bi,j. This means that matrices of the same size are added together by adding each element of the matrix to the corresponding element of the other matrix.Above, each matrix element is added to the corresponding element in the other matrix, which equals matrix C. Matrices of different sizes can not be added to each other.Subtraction of matrices are done the same way as adding matrices. Instead of adding each corresponding entry together, simply subtract them in the order they appear. This can also only be done when the matrices are the same size.Multiple matrices can also be added together by using multiplication when they are identical. For instance, Matrix D can be added to itself 3 times to have the sum of 3D. This would mean Matrix D is equal to D + D + D. The matrices are added together by multiplying each entry by the number of times it is added, which in this case is 3.Scalar multiplication of matrices are used in computer graphics. Each picture is input as a matrix, and the entries in a matrix are correlated with pixels. In a photo, each entry will generate a certain colored pixel. Photo effects are matrix operations. For example, by multiplying all of the entries of the matrix by -1, it reverses the signs of all of the entries in the matrix, resulting in a photo negative of the original. These are even done on iPhones, for people who have sensitivity to brightness, and makes it easier for people with low vision. An example of this is shown below.A row vector is a matrix with one row, and a column vector is a matrix with one column. The product of two vectors is the sum of the products of the corresponding elements. Let A and B be 1n or n 1 matrices. A has elements of a1, a2, …, an, and B has elements of b1, b2, …, bn. The inner product of A and B is shown using the formula below.A B = a1b1 + a2b2 + … + anbn An example of row vectors being used in real life is in business, when calculating the invoices and inventories. There is one vector for the amount of items, and another vector for the unit price of each item. Many stores use computer programs to calculate this information, and the creators of the computer program used vector multiplication.In order to multiply matrices, the matrices must be conformable. Conformability for multiplication is different than conformability for addition and subtraction. Two matrices are conformable if the number of columns for the first matrix is the same as the number of rows for the second matrix. The matrices A and B are conformable. This is because the numbers in red are equal, the amount of columns of the first matrix are equal to the amount of rows of the second matrix. The numbers in blue are the dimensions of the product matrix, which is matrix C, and it is a 3 by 3 matrix. When multiplying matrices, each row of one of the matrices will be multiplied to each of the columns in the other matrix, and vice versa. For example, row 2 in matrix A is being multiplied to column 2 in Matrix B, and has an element in matrix C as C2,2. If A is an m n matrix with rows A1, A2, …, Am, and B is an n p matrix with columns B1, B2, …, Bp, then the product AB is an m p matrix C whose elements cij are given by the formula   cij = Ai Bj.An example of multiplying matrices is shown below. Each number of the first row is multiplied with each number of the first column on the other matrix. This is done with every row and every column of the matrices.One of the most useful applications of matrices is the matrix representation of a system of linear equations. The linear system;The solution of this system (if it exists) is x = A-1b. It is difficult to accurately compute the inverse A-1, which remains a critical issue for computer analysts and mathematicians. Let A be an n n  square matrix. Matrix A-1 is an inverse of matrix A if it satisfies the two matrix equations A A-1 = I  and  A-1 A = I, where I is the identity matrix of size n n. An identity matrix is a square matrix if all the elements on its diagonal are equal to 1 and all the elements not on the diagonal are equal to 0. This is an example of an identity matrix.This solution is incredibly important to many fields of science that depend on calculating with large matrices. This includes astronomy, weather forecasting, statistics, economics, airline travel routes, investment banking, marketing studies, and medical research.Matrices are widely used in economics to manage databases, transportation and supply costs, international trade, surveys, production decision, inventory valuation, market equilibrium, and investment. One example of matrix multiplication is supply costs problem. For example, a building contractor is able to purchase the supplies lumber, bricks, concrete, glass, and paint from any of three suppliers. The prices that each supplier charges for each unit of these five materials are shown in matrix A.In matrix A, each row represents one supplier and each column represents the materials in the order listed. The contractor wants to buy all of the materials from one supplier for one job in order to lower transportation costs. There are currently three jobs; job one requires 20 units of lumber, 4 of bricks, 5 of concrete, 3 of glass, and 3 of paint. Job two requires 15, 0, 8, 8, and 2 units, of the same materials. Job three requires 30, 10, 20, 10, and 12 units, of the same materials. We need to find out which supplier he should use for each job to minimize the cost of materials. To solve this problem, we will arrange this information as a 53 matrix B and find the matrix product AB. In matrix B, each row represents the material units needed for each job, and the each column represents the number of the job.The product of multiplication of matrices A and B is matrix C with elements Cij where i is the supplier number and j is the job number. Cij represents the cost of the materials bought from supplier number i for job number j. In order to find which supplier the contractor should use for each job, we have to find the minimum number in each column. For Job 1, he should buy from supplier 1 for the lowest cost of $233, for Job 2, he should buy from supplier 1 for the lowest cost of $200, and for Job 3, he should buy from supplier 2 for the lowest cost of $490.Common geometric movements of figures, such as reflections and rotations, can be written as 22 matrices.Computer graphics use geometric matrices to model the changes of position of moving objects in space (such as space shuttle), transform them to eye coordinates, and project the three dimensional image onto the video screen. The matrix products must be calculated very rapidly to give the images realistic movements, so processors in high-end graphic computers insert the matrix operations in their circuits. The matrices used control the shuttle by using a rotation that makes the the nose go up and down, a rotation that causes the nose to rotate left and right, and a rotation that causes the shuttle to roll over. In graphic software, matrices used for transformations of images such as stretching or shrinking. The same matrix operations are built into medical instruments such as digital X-ray machines and MRI machines. The matrices used for these machines organize connections and distances between points. Airlines also use geometric matrices to find a profitable way in assigning planes to flights between different cities.Stochastic (random) matrices are formed from probability. An example of this is when probabilities are dependent on a prior state, the matrix will represent a Markov chain. For example, color shifts in generations of roses can be determined by having parts of a matrix connect with a set of probabilities that show a final state for the situation. Markov chains have been used to describe population growth, molecular genetics, pharmacology, tumor growth, and epidemics. Social scientists use matrices and Markov chains to explain voting behavior, populations of towns, and consumer choices. Albert Einstein used the Markov theory to study the motion of molecules, physicists have used it in the theory of radioactive transformations, and astronomers have used the theory to analyze the changes in the brightness of the galaxy. A picture of a Markov chain used in a matrix is shown below.Matrices provide a way of organizing information by using team results to determine unbiased ratings. Matrices can be used in situations where competing players have numerous choices of action. Each row can represent a player, and each column can represent an action. Game theorists have developed mathematical strategies for transforming the matrices in a way that gives the player the best outcome. Game theory strategies with matrices have also been used to determine which students get to be in college courses, resolve conflicts in classrooms, and economic behavior.Matrices are used for coding or encrypting messages. It can provide a more secure code than replacement ciphers. Replacement ciphers are used to code a message by replacing each letter with another one. This method makes it easy to decode secret messages. If a code is written with numbers that are put into code by multiplication of a matrix, the same letter is put into code with different letters, depending on its position on the coded message. When someone who has the encoding matrix receives the message, they are easily able to decode this message using multiplication with the inverse of the matrix.