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# Precalculus

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Matrices
Understanding and solving matrices.
All content in “Matrices”

### Basic matrix operations

Keanu Reeves' virtual world in the The Matrix (I guess we can call all three movies "The Matrices") have more in common with this tutorial than you might suspect. Matrices are ways of organizing numbers. They are used extensively in computer graphics, simulations and information processing in general. The super-intelligent artificial intelligences that created The Matrix for Keanu must have used many matrices in the process. This tutorial introduces you to what a matrix is and how we define some basic operations on them.

### Matrix multiplication

You know what a matrix is, how to add them and multiply them by a scalar. Now we'll define multiplying one matrix by another matrix. The process may seem bizarre at first (and maybe even a little longer than that), but there is a certain naturalness to the process. When you study more advanced linear algebra and computer science, it has tons of applications (computer graphics, simulations, etc.)

### Properties of matrix multiplication

As we'll see, some of the properties of scalar multiplication (like the distributive and associative properties) have analogs in matrix multiplication while some don't (the commutative property).

### Zero and identity matrices

In arithmetic, we learned than a number times 1 is still that number and that anything times 0 is 0. In this tutorial, we attempt to extend these ideas to the world of matrices!

### Geometric transformations with matrices

We'll now see one of the most powerful applications of matrix multiplication--geometric transformations. This is core of what videos games and computer-based animation uses to "transform" figures based on movement or perspective. You probably never thought matrices could be so much fun!

### Inverting matrices

Multiplying by the inverse of a matrix is the closest thing we have to matrix division. Like multiplying a regular number by its reciprocal to get 1, multiplying a matrix by its inverse gives us the identity matrix (1 could be thought of as the "identity scalar"). This tutorial will walk you through this sometimes involved process which will become bizarrely fun once you get the hang of it.

### Reduced row echelon form

You've probably already appreciated that there are many ways to solve a system of equations. Well, we'll introduce you to another one in this tutorial. Reduced row echelon form has us performing operations on matrices to get them in a form that helps us solve the system.