ACT Math Topic You NEED To Know In 2023 - Matrices & Matrix Multiplication
Summary
TLDRThis 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
📚 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.
🔢 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.
🧩 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.
Mindmap
Keywords
💡Matrix
💡Dimensions
💡Matrix Addition and Subtraction
💡Coefficient
💡Determinant
💡Matrix Multiplication
💡Defined vs. Undefined
💡Level of Difficulty
💡Practice Questions
💡Ultimate ACT Math Course
💡Free Trial
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
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