# 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).

Article

Intro to matrices

Matrix is an arrangement of numbers into rows and columns. Make your first introduction with matrices and learn about their dimensions and elements.

4:29

Intro to matrices

Take the red pill and enter the Matrix!

Exercise

Matrix dimensions

Determine the dimensions of a given matrix.

Exercise

Matrix elements

Find specific elements (also called entries) of given matrices.

## Representing linear systems of equations with augmented matrices

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

Article

Representing linear systems with matrices

Learn how systems of linear equations can be represented by augmented matrices.

Exercise

Represent linear systems with matrices

Determine the matrix that represents a given system of linear equations.

## 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.

Article

Matrix row operations

Learn how to perform the matrix elementary row operations. These operations will allow us to solve complicated linear systems with (relatively) little hassle!

Exercise

Matrix row operations

Determine the matrix that is the result of performing a specific row operation on a given matrix.

## Row-echelon form and Gaussian elimination

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

7:37

Solving linear systems with matrices

Sal solves a linear system with 3 variables by representing it with an augmented matrix and bringing the matrix to reduced row-echelon form.

## Adding and subtracting matrices

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

Article

Adding & subtracting matrices

Learn how to find the result of matrix addition and subtraction operations.

5:35

Adding & subtracting matrices

Sal defines what it means to add or subtract matrices. He shows a few examples and discusses some important properties of matrix addition and subtraction.

Exercise

Add & subtract matrices

Find the matrix that is the result of the addition or subtraction of two given matrices.

Exercise

Matrix equations: addition & subtraction

Solve equations where the unknown is a matrix, by using matrix addition and subtraction.

## 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.

Article

Multiplying matrices by scalars

Learn how to find the result of a matrix multiplied by a real number.

2:18

Multiplying matrices by scalars

Sal defines what it means to multiply a matrix by a scalar (in the world of matrices, a scalar is simply a regular number).

Exercise

Multiply matrices by scalars

Find the matrix that is a result of multiplying a given matrix by a given scalar.

Exercise

Matrix equations: scalar multiplication

Solve equations where the unknown is a matrix, by using matrix multiplication by a scalar.

## 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.

Article

Intro to zero matrices

Learn what a zero matrix is and how it relates to matrix addition, subtraction, and scalar multiplication.

Article

Properties of matrix addition

Learn about the properties of matrix addition (like the commutative property) and how they relate to real number addition.

Article

Properties of matrix scalar multiplication

Learn about the properties of matrix scalar multiplication (like the distributive property) and how they relate to real number multiplication.

## Multiplying matrices by matrices

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

Article

Multiplying matrices

Learn how to multiply a matrix by another matrix.

6:26

Intro to matrix multiplication

Sal explains what it means to multiply two matrices, and gives an example.

5:30

Multiplying matrices

Sal gives an example of a multiplication of two matrices that don't have the same dimensions.

Exercise

Multiply matrices

Find the result of a multiplication of two given 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!).

Article

Matrix multiplication dimensions

Learn about the conditions for matrix multiplication to be defined, and about the dimensions of the product of two matrices.

3:32

Defined matrix operations

Sal discusses the conditions of matrix dimensions for which addition or multiplication are defined.

Article

Intro to identity matrices

Learn what an identity matrix is and about its role in matrix multiplication.

7:59

Intro to identity matrix

Just as any number remains the same when multiplied by 1, any matrix remains the same when multiplied by the identity matrix. Learn more from Sal.

3:44

Dimensions of identity matrix

Sal explains why the identity matrix is always a square matrix, even though it works with non-square matrices.

Article

Properties of matrix multiplication

Learn about the properties of matrix multiplication (like the distributive property) and how they relate to real number addition.

7:32

Is matrix multiplication commutative?

Sal checks whether the commutative property applies for matrix multiplication. In other words, he checks whether for any two matrices A and B, A*B=B*A (the answer is NO, by the way).

8:37

Associative property of matrix multiplication

Sal shows that matrix multiplication is associative. Mathematically, this means that for any three matrices A, B, and C, (A*B)*C=A*(B*C).

4:42

Zero matrix & matrix multiplication

Just as any number multiplied by zero is zero, there is a zero matrix such that any matrix multiplied by it results in that zero matrix. Learn more from Sal.

3:30

Using properties of matrix operations

Sal determines which of a few optional matrix expressions is equivalent to the matrix expression A*B*C. This is done using what we know about the properties of matrix addition and multiplication.

3:58

Using identity & zero matrices

Sal solves a problem where he has to determine whether unknown matrices are zero or identity to make an equation true.

## Matrices as transformations

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

4:43

Transforming vectors using matrices

Sal transforms a 2-dimensional vector using a 2x2 matrix, and draws the original vector and its image on the plane.

Exercise

Transform vectors using matrices

Find the result (both algebraically and graphically) of transforming a vector using a transformation matrix.

7:15

Transforming polygons using matrices

Sal transforms a triangle using a 2x2 matrix.

Exercise

Transform polygons using matrices

Find the image of a given polygon under a transformation defined by a given matrix.

Article

Matrices as transformations

Learn how exactly 2x2 matrices act as transformations of the plane.

Article

Matrix from visual representation of transformation

Learn how to determine the transformation matrix that has a given effect that is described visually.

6:08

Visual representation of transformation from matrix

Sal finds the drawing that appropriately represents the effect of a given 2x2 transformation matrix on the plane.

Exercise

Matrices as transformations

Match transformation matrices with a visual representation of their effect on the plane.

## The determinant of a 2x2 matrix

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

1:10

Determinant of a 2x2 matrix

Sal shows how to find the determinant of a 2x2 matrix.

Exercise

Determinant of a 2x2 matrix

Find the determinant of a given 2x2 matrix.

## 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.

14:14

Intro to matrix inverses

Sal introduces the concept of an inverse matrix.

Exercise

Determine inverse matrices

Given a pair of matrices, determine whether they are inverses of each other.

14:27

Determining invertible matrices

Sal shows why a matrix is invertible if and only if its determinant is not 0.

Exercise

Determine invertibile matrices

Given a 2x2 matrix, determine whether it has an inverse.

## 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.

2:48

Finding inverse of a 2x2 matrix using determinant & adjugate

Sal gives an example of how to find the inverse of a given 2x2 matrix.

## 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.

Exercise

Find the inverse of a 2x2 matrix

Find the inverse matrix of a given 2x2 matrix.

## Determinants and inverses of large matrices

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

3:57

Determinant of a 3x3 matrix: standard method (1 of 2)

Sal shows the standard method for finding the determinant of a 3x3 matrix.

2:39

Determinant of a 3x3 matrix: shortcut method (2 of 2)

Sal shows a "shortcut" method for finding the determinant of a 3x3 matrix.

Exercise

Determinant of a 3x3 matrix

Find the determinant of a given 3x3 matrix.

13:36

Inverting a 3x3 matrix using Gaussian elimination

Sal explains how we can find the inverse of a 3x3 matrix using Gaussian elimination.

8:47

Inverting a 3x3 matrix using determinants Part 1: Matrix of minors and cofactor matrix

Sal shows how to find the inverse of a 3x3 matrix using its determinant. In Part 1 we learn how to find the matrix of minors of a 3x3 matrix and its cofactor matrix.

6:23

Inverting a 3x3 matrix using determinants Part 2: Adjugate matrix

Sal shows how to find the inverse of a 3x3 matrix using its determinant. In Part 2 we complete the process by finding the determinant of the matrix and its adjugate matrix.

Exercise

Inverse of a 3x3 matrix

Find the inverse of a given 3x3 matrix.

## 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.

9:55

Representing linear systems with matrix equations

Sal shows how a system of two linear equations can be represented with the equation A*x=b where A is the coefficient matrix, x is the variable vector, and b is the constant vector.

Exercise

Represent linear systems with matrix equations

Represent systems of two linear equations with matrix equations by determining A and b in the matrix equation A*x=b.

6:40

Solving linear systems with matrix equations

After he represented a system of equations with a single matrix equation, Sal solves that matrix equation using the inverse of the coefficient matrix.

14:20

Matrix word problem: vector combination

Sal finds the appropriate combination of two given vectors in order to obtain a third given vector. This is done by representing the problem with a single matrix equation and solving that equation.

## Model real-world situations with matrices

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

5:43

Matrix word problem: prices

Sal shows how matrices can be used to efficiently represent data about the prices of toilet paper and toothpaste in two different cities.