What are the applications of matrices in real world? Matrices are considered very important in mathematics. What are some examples of applications of matrices to real world problems that would be understandable by a layman?
 A: One would be hard pressed to find a tool in mathematics that is more widely used for real world applications than matrices and linear algebra.


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*Physicists and engineers to model physical systems and perform precision calculations needed for complex machinery to work.  Electronics networks, airplane and spacecraft, and chemical manufacturing all require fine tuned computations arising from matrix transformations.

*Routing problems and other problems in operations research make extensive use of very large matrices.  There are entire subfields of this discipline related to finding the fastest, most accurate solutions to various matrix problems, for purposes of managing very large supply chains.

*Many internet and computer programming companies also use matrices as data structures to track user information, perform search queries, and manage databases.  In the world of information security, many public key cryptosystems are designed to work with matrices over finite fields, in particular those which are designed with speed of decryption as a goal.
These are only a few examples, but let me assure you that nearly any quantitative line of work will make extensive use of matrix equations.
A: If you have system of nonlinear equations and linearize them, you get a linear system, which can be represented by a matrix.
So, matrices are relevant whenever you can model the real world with equations, which is very often.
A: A simple answer is matrix notation can be used as a way to solve a set of linear equations.  Linear equations are those that consist only of unknowns to the first power multiplied by constants, added together, and set equal to another constant.  
In the sciences, this may be used in chemistry to solve material and energy balance problems, electrical problems using Kirchoff’s loop current and node voltage laws, and mechanics problems using translational and rotational equilibrium equations.  
They may also be used in finance models to solve for supply and demand curve intersections and asset allocation problems.
These kinds of problems occur so frequently that most math software libraries and spreadsheets have built in routines to solve them if the number of unknowns is equal to the number of equations.  Look for “solving simultaneous equations’, ‘Gaussian Elimination’, and ‘Matrix Inversion’ in the documentation.  
This is such a handy approach that ways to extend the usefulness have been found.  For example, the constants in the first paragraph can be replaced by parameters of any mathematical form you have, so long as they do not contain any of the unknowns.  By a technique called Laplace transforms, systems of linear differential equations may be converted into systems of linear algebraic equations and again solved using matrix methods.  This allows the solution of some changing systems such as oscillating mechanical systems (pendulums, manometers), AC electrical problems, and process and mechanical control.  A matrix (transformation matricies) may be used to change the scale of an object, to model reflections across lines and points, find new or old locations after rotation of coordinate axes, and even do some simple derivatives in calculus.  A matrix and its inverse can be used together to move a vector during rotation of axes.  More applications can perhaps be found in linear algebra texts.  
But none of these answers may fit your everday life.  Consider how you felt about addition and subtraction when you were first learning them and now after you have become comfortable with them.  You now think about which one to apply anytime you have a problem in life involving the quantity of things.  It is the same with multiplication and division, fractions and decimals, roots and powers.  You may have actually come up with new ways of using them.  For matricies, can you make it fit the form in the first paragraph?  Make it a practice to consider how you can use any of your mathematical tools to situations you encounter and you find the best answer to your own question.
