# Matrices

Contents

Learn what matrices are and about their various uses: solving systems of equations, transforming shapes and vectors, and representing real-world situations. Learn how to add, subtract, and multiply matrices, and find the inverses of matrices.

## Introduction to matrices

Learn what matrices are and about their parts (coordinates and dimensions).

## Representing linear systems of equations with augmented matrices

Learn how systems of linear equations with multiple variables are represented with matrices.

## Elementary matrix row operations

Learn about the three basic operations we can perform on a matrix without changing the solution set of the linear system it represents.

## Row-echelon form and Gaussian elimination

Learn how to use the matrix row operations in order to easily solve large systems of linear equations.

## Adding and subtracting matrices

Learn how to add and subtract two matrices to obtain a new matrix (just like numbers!).

## Multiplying matrices by scalars

Learn how to multiply matrices by scalars (in the world of matrices, scalars are simply regular numbers). For example, multiply a 2X2 matrix by 3.

## Properties of matrix addition & scalar multiplication

Learn about zero matrices and about how matrix addition, subtraction, and scalar multiplication are related to their corresponding operations with real numbers.

## Multiplying matrices by matrices

Learn how to find the result of a multiplication of two matrices.

## Properties of matrix multiplication

Learn about identity matrices and about how matrix multiplication relates to real number multiplication (spoiler: the distributive property applies but the commutative property does not!).

## Matrices as transformations

Learn how matrices can be used as transformations of vectors or transformations of the plane.

## The determinant of a 2x2 matrix

Learn what the determinant of a matrix is and how to find it for 2x2 matrices.

## Introduction to matrix inverses

Learn what the inverse of a matrix is, and how to determine whether two matrices are inverses or whether a matrix is invertible.

## Finding the inverse of a matrix using its determinant

Learn about a way to find the inverse of a matrix using its determinant and adjuagates.

## Practice finding the inverses of 2x2 matrices

After you've learned how to find the inverse of a 2x2 matrix, gain some practice with it.

## Determinants and inverses of large matrices

Learn how to find the determinants and the inverses of matrices with dimensions 3x3 or larger.

## Solving equations with inverse matrices

Learn how to solve systems of linear equations using inverse matrices, and how to solve matrix equations using inverse matrices.

## Model real-world situations with matrices

Learn how matrices can be used to describe real-world situations.