ACT Math Topic You NEED To Know In 2023 - Matrices & Matrix Multiplication

Summary

This video tutorial offers a comprehensive guide to mastering matrices for the ACT Math section. It covers the basics of what a matrix is, dimensions, and how to perform addition and subtraction. The instructor then delves into more advanced topics, including finding the determinant of a 2x2 matrix and matrix multiplication, which is a high-level skill necessary for top-scoring students. The video also includes practice problems and offers free resources for further learning and mastery.

Takeaways

• 📚 The video is focused on teaching ACT Math, specifically about matrices, a topic many students find challenging.
• 🤔 Two main reasons for struggle with matrices are either never having learned about them or forgetting how to use them since freshman year.
• 📈 The instructor promises to cover everything needed to know about matrices for the ACT within the next 12 minutes.
• 📘 The video references 'The Complete Guide to ACT Math' as a comprehensive resource, with a link provided for interested viewers.
• 🔢 A matrix is introduced as a way to represent data, such as in the case of a system of linear equations.
• 📏 Matrix dimensions are explained as the number of rows by columns, which is important for understanding matrix operations.
• ➕➖ Matrix addition and subtraction are presented as simple arithmetic operations that require correct setup.
• 🔑 The video provides a tip on how to solve for individual elements in a matrix by setting up equations based on their positions.
• 🔍 The instructor emphasizes the importance of understanding matrix dimensions for easy questions and for more advanced rules like matrix multiplication.
• 📉 The video explains how to find the determinant of a 2x2 matrix, which is a key property used in certain ACT Math problems.
• 🔄 Matrix multiplication is identified as a high-level topic that is almost guaranteed to appear on the ACT and requires understanding of whether the matrices are defined for multiplication.
• 📝 The process of matrix multiplication is detailed, including how to determine if the operation is possible based on matrix dimensions and how to perform the calculation.

Q & A

What is the main topic of the video?

The main topic of the video is to teach ACT Math, specifically focusing on understanding and solving problems related to matrices.

Why do students often struggle with matrices?

Students struggle with matrices for two main reasons: either they have never learned about them before, or they have forgotten how to work with them since they were introduced in earlier years of study.

What is the purpose of the matrix in the context of the video?

In the context of the video, a matrix is used as a way to represent data, such as coefficients and constants in a system of linear equations.

What is the first level of difficulty mentioned in the video for ACT Math topics?

The first level of difficulty is considered beginner stuff, which should be easy for students to understand and solve.

What does the number 'one' in the matrix example signify?

In the matrix example, the 'one' signifies the coefficient in front of the variable x in the equation 10x - 3y = 40 + 1.

What are matrix dimensions and why are they important?

Matrix dimensions refer to the number of rows by columns in a matrix. They are important because they determine the size and structure of the matrix, which is crucial for operations like matrix multiplication.

How is matrix addition performed according to the video?

Matrix addition is performed by adding corresponding elements from the same position in each matrix. It's a simple arithmetic operation that requires the matrices to be of the same size.

What is the determinant of a 2x2 matrix and how is it calculated?

The determinant of a 2x2 matrix is a value that can be calculated using the formula ad - bc, where a, b, c, and d are the elements of the matrix arranged as [a, b; c, d].

What is the process for determining if matrix multiplication is defined?

To determine if matrix multiplication is defined, you need to check if the number of columns in the first matrix matches the number of rows in the second matrix. If they match, the multiplication is defined; if not, it is undefined.

How can one find the value of a variable in a matrix equation using the determinant?

By setting the determinant of the matrix equal to a given value, you can solve for the unknown variable by performing algebraic operations on the determinant equation.

What is the strategy for solving matrix multiplication problems on the ACT?

The strategy involves first determining if the multiplication is defined by checking the dimensions, then performing the multiplication by taking corresponding elements from the rows of the first matrix and columns of the second matrix, multiplying them, and summing the products to fill in the resulting matrix.

How can one check the answer choices after solving for one spot in a matrix?

After solving for one spot in a matrix, you can check the answer choices to see if any can be eliminated based on that spot's value. If multiple choices match, you'll need to solve for another spot to determine the correct answer.

What is the significance of the video offering a free trial to the ultimate ACT course?

The free trial to the ultimate ACT course allows students to access additional resources, including more practice questions and video explanations, to further enhance their understanding of matrices and other ACT Math topics.

Outlines

00:00

📚 Introduction to ACT Math Matrices

The video script begins with an introduction to the topic of matrices in ACT Math, addressing common struggles students face with this subject. The instructor promises to teach everything needed to know about matrices for the ACT within the next 12 minutes. The script mentions a book, 'The Complete Guide to ACT Math,' which is recommended for further study. The instructor explains the basic concept of a matrix as a way to represent data, using a system of linear equations as an example. The script also introduces the four levels of difficulty in the ACT Math book, with matrices being a level one topic, and touches on matrix dimensions, which are crucial for understanding matrix operations.

05:01

🔢 Understanding Matrix Operations: Addition, Subtraction, and Scaling

This paragraph delves into the basics of matrix operations, specifically addition and subtraction, which are presented as simple arithmetic problems. The script provides an example of how to perform these operations, emphasizing the importance of staying in the correct position within the matrix. It also discusses the concept of matrix dimensions and how they relate to the number of rows and columns. The instructor explains that these dimensions are essential for determining the validity of matrix multiplication. Additionally, the script introduces the concept of scaling matrices by distributing coefficients across the matrix elements, using an example to illustrate the process.

10:01

🧩 Advanced Matrix Topics: Determinants and Multiplication

The script moves on to more advanced matrix topics, starting with the determinant of a two-by-two matrix, which is a property that can be calculated using a simple formula. The instructor demonstrates how to calculate the determinant and use it to solve equations, as shown in an example problem. Matrix multiplication is the next topic, which is considered a higher-level concept and a common feature in ACT Math tests. The script outlines the steps to determine if matrix multiplication is possible based on the dimensions of the matrices involved. It also explains the process of matrix multiplication, emphasizing the need to align corresponding elements from the two matrices and perform the necessary multiplications and additions to fill in the resulting matrix.

📝 Applying Matrix Concepts with Practice Questions

The final paragraph of the script encourages viewers to apply the concepts learned by attempting four practice questions of varying difficulty levels. The instructor offers resources such as a free trial to an Ultimate ACT Math course and downloadable materials, including the entire chapter on matrices with additional practice questions and video explanations. The script concludes with an invitation for viewers to ask questions or request topics for future videos, emphasizing the instructor's willingness to assist with any ACT Math inquiries.

Keywords

💡Matrix

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. In the context of the video, matrices are used to represent data and equations, which is fundamental for understanding the ACT Math section. The script uses the example of a system of linear equations that can be represented as a matrix, illustrating how matrices can simplify the process of solving such equations.

💡Dimensions

Dimensions in the context of matrices refer to the number of rows and columns a matrix has, often expressed as 'rows by columns'. The script explains that understanding matrix dimensions is crucial for performing operations like addition, subtraction, and multiplication, as it dictates the possible outcomes and the structure of the resulting matrix.

💡Matrix Addition and Subtraction

Matrix addition and subtraction are operations performed on two matrices of the same dimensions. The script emphasizes that these operations are straightforward, involving the addition or subtraction of corresponding elements from the matrices. It is an essential concept for the ACT Math section, as it simplifies the process of combining or comparing systems of equations represented as matrices.

💡Coefficient

A coefficient in mathematics is a numerical factor multiplied by a variable or term. In the video script, coefficients are mentioned in the context of multiplying matrices by a scalar value, which is a common operation in matrix algebra and can be encountered in ACT Math problems involving matrices.

💡Determinant

The determinant is a scalar value that can be computed from the elements of a square matrix. It is used for various purposes, including determining if a matrix is invertible. The script explains how to calculate the determinant of a 2x2 matrix, which is a key concept for understanding certain properties of matrices and solving related ACT Math problems.

💡Matrix Multiplication

Matrix multiplication is an operation that takes a pair of matrices and produces a new matrix. The script outlines the rules for determining whether two matrices can be multiplied and how to perform the multiplication if they can. This concept is crucial for the ACT Math section, as it is a common operation that can be used to solve complex problems.

💡Defined vs. Undefined

In the context of matrix multiplication, a matrix operation is 'defined' if the number of columns in the first matrix matches the number of rows in the second matrix, allowing for multiplication. If not, the operation is 'undefined'. The script explains this concept, which is essential for understanding when matrix multiplication is possible.

💡Level of Difficulty

The script categorizes matrix-related topics into different levels of difficulty, ranging from beginner to expert. This classification helps students gauge their understanding and prepare for the varying complexity of matrix-related questions on the ACT Math test.

💡Practice Questions

Practice questions are exercises provided in the script to help students apply and reinforce their understanding of matrices. The video mentions different levels of practice questions, indicating the importance of practicing a range of problems to prepare for the ACT Math section.

💡Ultimate ACT Math Course

The Ultimate ACT Math Course is a resource mentioned in the script that offers in-depth lessons, practice questions, and free trials to help students master the ACT Math section. It represents an external resource that students can use to improve their skills in areas such as matrix operations.

💡Free Trial

A free trial is an opportunity for students to access a limited version of a course or service without cost, as mentioned in the script for the Ultimate ACT Math Course. This allows students to explore the course material and decide if it's suitable for their needs before making a commitment.

Highlights

Introduction to ACT Math matrices, a common topic that students struggle with.

Explanation of matrices as a way to represent data in a structured format.

Clarification of matrix dimensions, described as height by width.

Matrix addition and subtraction are introduced as simple arithmetic operations.

Demonstration of solving for matrix elements using basic algebraic methods.

The importance of understanding matrix dimensions for easy question solving.

Introduction of coefficients in matrix operations and how to distribute them.

Finding the determinant of a 2x2 matrix as a property of the matrix.

Explanation of the determinant formula for 2x2 matrices (ad - bc).

Solving for variables using determinant equations in matrix problems.

Matrix multiplication as an almost guaranteed topic in the ACT Math section.

How to determine if matrix multiplication is defined based on dimensions.

Step-by-step guide on performing matrix multiplication.

The process of multiplying matrix elements and adding the products.

Praxis problems with varying difficulty levels for practice.

Offer of a free trial to the Ultimate ACT Math course for additional resources.

Availability of the complete chapter on matrices for free download with video explanations.

Invitation for viewers to ask questions about the ACT or other math topics for future videos.

Transcripts

in today's video we're going to do some

ACT Math and we're going to learn

matrices now Matrix are one of the most

common topics I see students struggle

with for two reasons number one you may

have never learned matrices at all so if

you don't know what this is You're Not

Alone number two you may have learned

these back in like freshman year and

totally forgotten how to do them so if

you stick with me for the next 12

minutes I'm going to teach you

everything you know about matrices for

the act so be able to answer any

matrices question you see on test day

correctly now you can see on the screen

here is chapter 19 from my book the

complete guide to act math it is

absolutely the best AST math book out

there I'll post a link in the

description down below if you want to

grab a copy we're going to use this

chapter for me to teach you guys

everything you know about matrices now

let's start with what is a matrix well a

matrix is just a way to represent data

so I can have a matrix representing say

this right here I have my 10x minus 3y

equals 40. 4X plus 8y equals 18. I could

just represent this as a matrix over

here and we can recognize is the First

Column the 10 and the four are just

showing us the X values the negative 3

and the eight are showing us the Y

values and the 40 and the 18 are showing

us the constants so you might look at

that and go okay an AC question asked me

to do that would be super super easy and

that explains why we have a one right

here so what the one shows us here is in

my AST math book there are four levels

basically one is like the beginner stuff

two is the more intermediate three is

the more challenging four is like the

expert level stuff and I'll put a little

um description on the screen of the

different levels in your different

scoring goals so this is a level one

topic between ever want to be able to do

this now let's go to the second thing we

need to know matrices which is matrix

dimensions Dimensions you can think of

as height by width technically it's the

number of rows by columns this will be

helped for a couple things with matrices

here's an example this is a two by one

just think two tall one across this here

is a two by two two tall two across or

two rows two columns this is a one by

three matrix dimensions come up on easy

questions it'll more commonly come up

with one of our more advanced rules

we'll see later in this video with

matrix multiplication now the other easy

topic we'll see early on the test that

stumps a lot of students is Matrix

addition and subtraction and Matrix

distance of traction is actually really

really easy it is just simple arithmetic

like a second or third grader could do

this but they just need to know how to

set it up so here if we're doing Matrix

Edition all we need to do is stay in the

same spot so if I'm doing this one just

look top left negative five

plus 10 well if we add negative 5 and 10

we get 5 let's do top right four

plus seven well four plus seven is

eleven bottom left stay bottom left

bottom right say bottom right you

probably get it so we see a question

like this in the test like example one

down here this is an easy question all

you need to do is just do the math stay

in the same spot so we're trying to find

a plus b well

if I want to find a that's top right so

let's set up our top right equation

14 plus a equals 9 to solve for a

subtract 14 from both sides

we'll get a equals negative 5. all right

what about B well if we want to find B

we can basically look at b as bottom

left so let's just set up our bottom

left so we have negative 9 plus 6 equals

B if I add those together we find that b

is equal to negative 3 and you're asked

to find a plus b so negative 5 plus

negative 3 gives us negative 8 and the

answer is e so super super simple the

only way they can make this harder on

the ACT is by putting a coefficient in

front like we see in example two and if

we see coefficients in front of a matrix

all we do is distribute those values in

so I'll go ahead and solve example two

and show you how this works so in

example two it says we have three times

the first Matrix minus four times the

second Matrix and we're trying to find

what it equals so all we're going to do

here is distribute the 3. so if I'm

going to solve for the top left of my

matrix it's going to be 3 times 2 is 6

and then negative 4 times 5 is going to

be pi positive 20. so the top left is

going to be 6 Plus 20. and a pro trip

tip here is once you solve for one spot

in The Matrix we see the top left is

going to be 26 check the answer choices

a lot of times you can solve for just

one spot and find the answer here I can

get rid of C D and E both A and B have

26 so we have to solve for a different

spot we see say the top right is

different so let's solve for top right

so we're in the top right 3 times

negative 10 is negative 30 and then

minus 4 times negative 8 is going to be

plus 32. if we do the math there

negative 30 plus 32 is going to give me

2 in the top right and we can say boom

the answer is B so again this is the

really really easy matrices questions

you'll see these in the first like 20 or

30. I think everyone no matter your math

level should be able to solve these

pretty easily next let's talk about our

Advanced matrices topics Beyond addition

subtraction make sure you usually come

up at the end of the AST and those more

difficult questions from say 40 to 60

especially from 45 to 60 towards the end

of the test now one of these topics is

actually easy I just called a level two

here because again as long as you

memorize our equation it's pretty simple

you see it in a second so this is called

find the determinant of a two by two

Matrix don't worry what that about what

that fancy word is it's just a uh it's

like a property of a matrix so here all

I need to remember is with a two by two

Matrix the determinant if it's Matrix a

b c d is a D minus BC so just top left

times bottom right minus top right times

bottom left so if I have this one here

the determinant is just a d 2 times 12

minus BC negative 6 times 3 if we just

do the math here 2 times 12 is 24 minus

negative 18 oops negative 18.

turns into plus which gives us a value

of 42. so again you look at a question

like that you go oh that actually be

pretty easy so if we see one like

example three here this again even

though it's

um not a hard question it's actually

going to appear towards the back of the

test so just make sure you memorize that

equation so the determinant of Matrix B

is equal to 10 what's the value of x

well we just got to set up our

determinant equation so the determinant

of this is top left times bottom right

so it's negative 4X minus top right

times bottom left well 1 times 6 is 6.

it tells us that equals 10. well this

looks pretty easy to solve so all we

need to do is add six

so we get Negative 4X equals 16. we can

divide both sides by negative 4 and we

get x equals negative 4. boom we're done

even if this question like 45 or 50 you

can solve this pretty easily just again

memorize that little a D minus BC

equation

next let's talk matrix multiplication

now this is almost guaranteed to be in

your act it's on almost every single one

it's at the back of the test somewhere

in questions 40 45 to 60. now as you can

see here this is a level three to four

topic it's one I want students who are

aiming for anything 28 29 30 and up to

make sure you really understand because

again you're almost guaranteed to see

this now if you understand how it works

which I'll show you in a second it's

actually not that bad now our first step

with matrix multiplication is always

figuring out whether matrices are

defined versus undefined and the steps

are written here I'll go through them

quickly so let's say I have a times b

and a is a one by two Matrix and B is a

two by three Matrix to see if these are

going to be basically possible to

multiply we just write down the

dimensions so here if I'm doing a times

B it's 1 by 2 and 2 by 3. what we then

should do is we're going to check the

middle number so make a box around those

middle numbers if those middle numbers

match then the Matrix is going to be

defined and what we do is you can kind

of see down here once I've already made

the box if it matches we're going to

take down those outside values and we're

going to get year A 1 by 3 Matrix now if

I instead did Matrix B times a then I

would write Matrix B first which is a

two by three I'd write a second which is

a one by two and we would see that these

middle numbers don't match if the middle

numbers don't match that means B times a

is going to be

undefined so we may see some questions

like example four right here in the test

that just ask you to understand this

step so here we're using the matrices

below we're trying to find which is a

two by three Matrix well a is going to

be a one by two

B is going to be a two by two c is going

to be a two by three so think about

setting that up which one's going to

give us a two by three Matrix well here

it's going to be BC the reason is if I

set up B I'll write it here BC B is a

two by two Matrix C is a two by three

Matrix if we do our box that's going to

work so now we bring down our outside

values and we can see BC is going to be

a two by three Matrix now to finish up

here let's talk about how to actually do

the matrix multiplication which is the

most common way this is tested on the

act so here if I need to do the actual

multiplication let's say I'm going to

use these two matrices right here I have

a which is a two by two Matrix and I

have B which is a two by two Matrix our

first step in matrix multiplication is

always what we just learned we have to

figure out is it going to be defined or

undefined and what are the dimensions

going to be well here as we can see a

two by two Matrix a is two by two times

a two by two is going to match in the

middle and we're going to bring it down

and get a 2 by 2 meters so that's kind

of the easy part with this one but

knowing those Dimensions could be

helpful to eliminate amps Choice a lot

of times when you're doing these on the

test now what we're going to do from

here is we're going to work horizontally

across our first Matrix Matrix a and

vertically down our second Matrix Matrix

B and the way I like to teach this is I

have to imagine let's say this is our

Matrix we're solving for here

it's a two by two Matrix so we have four

spots we're going to be solving for now

when are we going to solve for the top

left well we have to work horizontally

across the first Matrix vertically down

the second so imagine if I'm drawing a

horizontal line excuse me horizontal

line and a vertical line

where are those going to cross well here

it's going to cross top left so that's

why we want to work across the top and

down the left hand side as you can see

by those shaded values here now we're

going to do is we're going to start left

and start top so I'm going to start at

the left with the 2 the top with the

three and we're going to take our first

pair of numbers and multiply them 2

times 3 is 6 and then we're going to

shift right and shift down to our second

set of numbers our second set of numbers

is the negative 4 and the negative 2 we

multiply those together that gives us

positive eight we add those together and

that gives us the value that we're going

to put on the top left so that's why we

see right here we put a 14 on the top

left part of the Matrix so it's kind of

this weird combination of like

multiplication and addition now if I

want to solve for top right now we have

to go across the top but down the right

hand side so you can see here we're

across the top but now down the right

hand side and I'm going to basically use

that same exact technique starting left

starting top 2 times 1 is 2 then we're

going to shift to the right we're going

to shift down we're going to multiply

our second set negative 4 times 9 is

going to be negative 30.

six we add those together we get

Negative 34 which is why the negative 34

goes in our spot on the top right you

probably see the pattern already I'll

kind of show you how the last two work

just to be super thorough here but

usually in the test once you find one of

these spots again go check the inch

choices you can often just do the first

step we did up here and already tell

what the answer is so on your own go

ahead and see if you can find this

bottom left point and then I'll show you

right now how to do it so if you want to

pause it and take a shot at it go for it

to the bottom left here what we want to

do is we want to go across the bottom

and down the left and again think if I

redraw my Matrix we can see it across

the bottom down the left is going to

solve us for that bottom left spot so

we're going to take our values we're

going to start with negative 6 times 3

is going to be our negative 18. we're

going to do our 1 times 2 which will be

plus 2 sorry plus negative two try it

again and that gives us a value of

negative 20 in the bottom left if you

want to solve for the bottom right as

well I'll just show you the bottom right

value would be 3 so this is going to be

our entire Matrix and solve that so

again if you can handle this you'll be

prepared to solve the most difficult in

each of these questions you're going to

see on the ACT

now let's try and apply what you've

learned so you see here are four

questions from the Praxis problems in

this chapter so we have one level one

one level two one level three one level

four so go ahead and pause the video

grab a pencil piece of paper try and

work through these four in a second I'll

show you what the answers are and also

show you how you can watch videos of me

explaining all these questions in many

more practice ones completely for free

all right so now I can see the answers

to these four practice questions on the

screen if you want to watch videos of me

explaining how to do each one of these

questions you can get that in the free

trial to my ultimate HT course in

addition you can actually download the

entire chapter 19 about matrices for

free there's 19 more practice questions

and again there's videos of me

explaining how to do every single one of

those practice questions so it's a great

resource to help you master matrices in

addition if you guys want to learn more

math in the free trial to my ACT Math

course there's three more actual

chapters you guys can do you can get

downloads of it do um the entire watch

the lessons do a bunch practice

questions so again it's all resources

that are totally for free no credit card

needed that should help you guys improve

on the HDM so there's links to all that

stuff below there's also a link if you

want to grab a copy of my book on Amazon

other than that if you guys have any

questions about Annie made she's related

or if you have other math topics you'd

like to see in future videos please let

me know if you have any AC questions in

general I'm happy to take time to answer

them so again fire off in the comments

below other than that this is not a Prep

Pro signing off I will see you guys next

time