1 - Intro To Matrix Math (Matrix Algebra Tutor) - Learn how to Calculate with Matrices

hi and welcome to the matrix algebra tutor and in this class we're going to focus on a part of algebra that is sort of it sort of compartmentalized a lot of times you'll learn about this stuff in college algebra you'll learn about it sometimes at the very end of an algebra 2 class depending on your class and of course when you're off doing engineering and other things you'll you'll have a dedicated class in this called linear algebra at least some of the topics in your linear algebra class will be covered by by everything on this DVD and additionally more things as well okay so the point is matrix algebra which we're going to learn about over the next several hours is very integral and it's it's a part of algebra that's very very useful and it's a part of algebra that can throw people for a loop mainly because it looks completely different than anything you've ever learned in algebra once you start dealing with it I mean in algebra I mean it does have the word algebra in the title so you expect to see a lot of X's and Y's and exponents and things like this but matrix algebra or matrices they usually don't have any variables at all and I think you might have a fair idea of what a matrix is just by thumbing through your textbook you you see these square brackets and and you know you know that there's some sort of math going on here but it looks completely different than anything else you've probably learned up until that point and so it looks sort of like gibberish and it looks like it's gonna be difficult but the fact of the matter is once you take it one step at a time from the beginning and march through what these things are about and work worried about how to how to use them and how to manipulate them just like you had to learn how to manipulate your variables and your Exponential's and everything else we're gonna learn about how to manipulate these matrix matrix things okay and it's not going to be a big deal that's the point so this section of the class is the first section in what we're gonna learn about mainly is what what is a matrix okay let's write it down let's talk about its components let's get comfortable looking at them we're not gonna be adding anything or subtracting anything we're just going to learn about what a matrix is and why they're useful okay so let me start with the first thing the motivation why do we care about matrix algebra why do we make it a point to learn matrix algebra okay well let me ask you this when you remember back to your algebra a long time ago at some point you did study something called solving a system of equations that's what you you studied it's covered in my DVDs it's covered in every algebra class that you're gonna take whether it's college algebra or algebra 2 and high school or whatever you will study something called solving a system of equations that's a big fancy thing and all it means is if you remember back you have instead of just one equation to solve you know like X you know plus 2 is equal to 9 that would be one equation a system of equations is usually more than one equation in more than one variable so for instance you might have instead of one equation you might have two separate equations and because you have two equations you're trying to solve for two unknowns okay x and y let's say or if you want to have a more complicated problem you could have three variables you're trying to solve XY and z and because you have three variables you must have three equations in order to solve for all the variables so the short and long and short of it is you have to have as many quai shion's to describe the system you're talking about as variables you're trying to solve so as a practical example let's say in this room you know this gas this air around my head right now okay it has a certain temperature okay it has a certain pressure you know the pressure goes up and down with the weather depending on the cold fronts and warm fronts that come through and it might have a certain humidity let's say so those are three variables we could call them x y&z whoo and you know if you're solving a real problem you might actually have a problem that deals with the pressure of some gas the temperature of some gas and let's just say the humidity of the gas so three variables and you might have three equations in order to solve those three variables so what you would do you would write down the equations that describe you know the system and you'd have to have three different equations to solve for those three variables if that's what you were trying to do and you were taught several different ways to do that you were taught if you remember back from algebra you were taught how to solve that by substitution you can take one of those equations you can solve it for a variable plug it back in another equation and by doing a lot of substitution over and over again you can eventually solve for one of these variables and continue solving for the remaining variables so you can use substitution you also probably learn something called addition solving them by addition or sometimes it's called solving them by subtraction basically that is just taking the equations and lining them up one under another and you can you can long as short of it as you can subtract them from one another and by doing that you end up eliminating some variables and solving for some variables okay so what am i leading to why why am I taking you down this memory lane of algebra that you've studied you know before because solving a system of equations which is just simply more than one equation and of course more than one variable because you have more than one equation okay you already know a few techniques to solve those things you've already done that before the main use of matrices okay the main use of matrices or matrix algebra is doing that solving the system of equations in a shorthand fast way that's really what it is okay so if you have to take one thing away from this section at least in the beginning here the entire point of this entire DVD course is not just to bore you with a bunch of matric matrices on the board it's because it has a point and the point is you can use this matrix stuff to solve systems of equations that are very useful to learn how to solve and real life problems okay and you can do it without all that substitution without all of that equation addition you're going to use it solve it using matrices and you're gonna find out that once you figure out how to crank through it it's going to be actually not too bad and it's gonna save you time because just gonna save you from writing down a lot of other steps if you remember back solving those systems it took a fair amount of paper to do because you're writing all the equations down every time and solving them with a matrix once we show you how it's not going to be a big deal okay so that's where we're going that is the end game the first few sections of this course are gonna teach you what a matrix is how to add them subtract them and everything else sort of leading up giving you the skill so once we get to the middle of the class here in the DVD you'll learn how to actually solve systems of equations and understand what these things are for now matrices have uses far beyond that but that is certainly the most important thing for you to remember now and certainly probably the most practical reason that we use them in computers use matrices to solve these these things all the time if you're gonna you know if you're going to write a computer program to solve a real problem for more than one equation you're definitely going to use a matrix to solve it no question about that okay so let's finally dive into the topic here and figure out what a matrix is okay the the big deal here is a matrix is nothing more than some rows and columns of numbers okay for a minute even though I've kind of given you the endgame of where we're going that we're going to use these matrices to represent system of equations just forget about that for now just put it in the back of your head and remember that's what we're going to use these things for but for now just open your mind and and and learn about what a matrix is and it's simply a square bracket with some numbers inside okay it's not that big of a deal so let's write that down okay a matrix contains rows and columns of numbers now notice I didn't say that it contains variables I didn't say that it contains exponents I said it's rows and columns of numbers so when you think about that even though they might look a little funny at first and we'll get to that here in a second it's really just some brackets filled with numbers and you've been dealing with numbers ever since you were a kid so you know even though it looks foreign to you at first it's not that bad there's not a bunch of variables running around here for you to keep track of with a matrix usually you're gonna be dealing with numbers and numbers are much easier to wrap your brain around usually than a foreign concept with variables okay so what what is a matrix the easiest way to do it is just to write one down okay here is what a simple matrix would look like you put a square bracket and there's some numbers inside let's put one two three and four okay congratulations we've just written your first matrix down okay now the first thing most people want to do when they first learn about matrices is you'll see these numbers here and it's confusing there's no addition subtraction division there's there's no operation in here so it looks a little bit odd there's no equal sign anywhere okay and it just it's a it's a bunch of numbers but it just doesn't look like it's very useful okay and what you have to remember is just like when you first started algebra you have to learn the basics of this stuff in order to be able to use use it for what it's really intended okay just like you had to learn what a negative number was and that was a really complicated thing way back in the day well you're gonna have to learn what these things look like okay so these things are called matrices and notice that there's some rows there's two rows in this matrix and there's two columns in this matrix okay so you see here that you have two rows in this matrix and you have here two columns okay whoops gotta spell columns right okay two columns 2 rows and 2 columns why is that important because you'll see later on that when we write different types of matrices you can have different numbers of rows and different numbers of columns so generally when you're writing these things down you refer to the order of the matrices or the order of the matrix just to describe how big it is what its shape is okay so what you would say in this case is that the order of this matrix which you'll probably have to do in your test you'll be given a matrix and it'll say what's the order of this matrix and you'll be like what is that well you have to learn the definition the order basically just says how many rows how many columns okay the first number is the the rows and the second number is the columns okay so this is 2x2 the X's is not a variable X its it's like like a 2 by 4 piece of lumber or something that's what the X means here so it's a 2 by 2 matrix two rows and two columns so you see this is not this is not rocket science okay there is no variables here there's nothing complicated here there's some numbers inside of a bracket now the only thing you might beat your head against the wall trying to figure out is what would this possibly be used for I've already given you the punch line you see this matrix later on we're going to learn this matrix can be used to represent an equation actually it can be used to represent a system of equations just like you were solving those systems of equations when you first learned algebra right and we're gonna use these matrices to solve them easier to basically solve them easier and that's really what it's all about okay so this is a 2x2 matrix it's called that's called the order of the main if you have a question on your test what's the order of a matrix you write down how many rows how many columns put an X in the middle and you read it 2 by 2 or 3 by 2 or 5 by 4 or whatever that's the order of the matrix okay so let's go ahead and write down what the order of a few other matrices would be just to get a little bit of practice okay let's say we have a matrix that looks like this zero negative 1 3 0 1 and 4 notice I do have a negative number in here notice I do have some zeros in here so I said the matrix is usually gonna be full of numbers these numbers can be negative positive they can be decimals they can be fractions it can be any number that you want really any any real number that's what we're gonna use these things for okay the 0z here we still have to put them in their place okay and again this would represent I'm giving you the punch line a little bit later on we'll see that this represents a system of equations okay that that you would then use this matrix to solve okay so that's where we're going with us if you were trying to find the order of this matrix all you got to do is look and say well look it has two rows two rows by three columns one two three columns so it's a two by three order two by three okay so that's another matrix and just to give you a little bit more practice let's go ahead and write another one let's switch colors a little bit what if I have notice I'm kind of drawing a kind of a vertical skyscraper matrix here one three let's say seventy four just for kicks twenty nine let's say negative five and negative 1 okay so we've got some big numbers we've got some negative numbers here but again it's the same thing what is the order of this matrix if you were asked that on your test you would say well how many rows do I have I have three rows so that number goes first and two columns so it's a three by two order okay so we haven't we haven't really done anything with these things yet we've just talked about the fact that we have this thing called an order which is the number of rows and the number of columns okay and that's sort of just a definition more than anything else we haven't done any operations with these things we're just writing them you have no idea at this point how you would use this thing to solve a system of equations but that's fine because we will get there okay we'll get there now let's continue learning about a matrix and learn about the elements of a matrix okay that element sounds like a complicated thing but really when you look at this matrix here it has numbers inside and each number inside of this thing is called an element of the matrix okay so this thing's this entire entity is called a matrix and everything inside of it every single number is called an element of the matrix so if you're asked to find a certain element or pull out a certain element of a matrix to identify it just on a test to see if you know what you're talking about you know that you're gonna be looking inside this thing try to pull a number out that's what that's what you're gonna be doing okay so the elements of a matrix the elements of a matrix are the numbers inside okay and numbers inside okay so let's go ahead and draw that the easiest way to do this is by example okay let's say you have let's say well let me start it back over a little bit here and say that usually in your textbook you will see the elements of a matrix written as the letter A usually not always with the subscript I and a subscript J okay now this is usually when people start to get confused about this matrix stuff because you have some variable and you have two little subscripts what does all this stuff mean okay this is just used to identify what element of the matrix you're talking about and it's very very simple let me show you what I'm talking about here by an example let's say we have a matrix one two three four five six seven eight nine before we go any further what would the order of this matrix be okay just think about that but you have to look at the rows and the columns there's three rows and there's three columns so it's a three by three 3 X 3 it's a three by three order matrix ok now the element of the matrix are identified by a usually the books use a with a subscript I and a subscript J don't get worried about this okay all it means is this number the first one the first subscript identifies the row that you're talking about and this number identifies the column that you're talking about and once you've identified the row in the column that you're talking about you can identify any number uniquely in here okay so all you're doing is to pull out an element or to identify an element of a matrix you're gonna generally have a some represent some variable really just to show that you're talking about an element of a matrix and you're gonna have two little subscripts the first subscripts gonna identify the row you're talking about is that this row is it this row or is it this row the second number is going to talk about the column that you're referring to in the intersection of those two things is the element you're talking about so to put it in more concrete terms let's practice okay let's say I'm trying to find element a sub 1 1 on a test you're given this matrix and you're told or asked what is the element denoted by a 1 1 so all you have to do is realize the first number is the row so it's the first row this is Row 1 Row 2 and Row 3 this is column 1 column 2 in column 3 so you're in row number 1 up here ok now in order to find out what element you need which is what number you need you're in column number 1 so this is column number 1 this is row number 1 so this guy right here is the element you're talking about so you see how easy this is you just have to know how to interpret it you look at the row you look at the column that's the element you're talking about because frequently when you're dealing with a matrix you might have to refer to the elements of the matrix you might have a need to have write an equation involving the elements of the matrix so this little subscript notation is how you do that it's like a crossword puzzle I mean really it's no different than a crossword puzzle you know three across two down I mean it's really all it is row and column that's how you identify the elements of the thing okay so what if you were trying to find a 1 3 ok a 1 3 first number is the row so I'm on row number 1 second number is the column column number three so the element is three okay let's say I had to find a two - okay a two - and by the way when I say a here a here and a here that's if I'm if I'm calling this matrix in the matrix a okay I could be calling it that labeling it matrix a then these little subscripts denote the elements of the matrix a ok that's why if you were gonna call this this matrix you know matrix B or a matrix Z or matrix G then you would have G with the subscripts 1 2 1 3 or whatever depending on what you actually labeled the thing we're just using a because it's it's easy okay so Row 2 column 2 here's Row 2 here's column 2 intersection is 5 ok a 2 3 Row 2 column 3 here's row number 2 here's column number 3 that's a 6 ok and then let's say a 31 I shouldn't say a 31 it's a 3-1 it's not really the number you care about it's it's the individual digits here so Row 3 over here and column number 1 intersection is 7 ok and finally let's say I'm going to find a 3-2 so I look the first digit is the row the second digit is the column so the first digit is Row 3 so I'm in Row 3 column number 2 this is column number 2 the intersection that's 8 so the answer is 8 so we can't come up with one three five six seven and eight and and basically you can do this for any element of the matrix if the matrix had let's say this matrix was huge this matrix could have you know a hundred rows and 519 columns I don't know why you would ever need to create a matrix that big but you could if you wanted to and so in order to to to look at the individual elements of that matrix the individual number so all you need to do is know as two numbers the row it's on and the column it's on once you have that you're in good shape so this is a good introduction to matrices here in the rest of this section we're going to talk a little bit more about when matrices are equal when when when the different matrices are equal and and just get a little bit more practice with identifying the elements of the mate okay now something else that's very important to understand about matrices we have drawn actually several matrices so far some of them have the same number of rows and columns and so they're called square matrix I don't even think I mentioned that before but that's something that you probably need to remember when it has the same number of rows and columns it's gonna look like a square that's called a square matrix okay if it has different number of rows and columns which can always be the case and it's a non square matrix okay that makes sense because it looks more like a rectangle right and the thing that you need to remember is if you have two matrices and the test or some book asks you are these matrices equal are they equal basically all you have to remember is a matrix is equal to another matrix if and only if they had exactly the same shape and size in other words same number of rows same number of columns and every single element inside those matrices are equal to the corresponding element in the other matrices so basically they have to look exactly identical in all respects the size the shape and exactly every little element inside they have to match exactly otherwise they are not equal okay so let's write that down and get a little practice so we say that two matrices are equal if they have the same size and shape and all elements are equal now notice here I wrote something down they have the same size and shape because you know that to me is everyday language everybody's gonna understand okay basically what this means and you'll see this a lot in your book let me just go ahead and put a dotted line around it same size and shape this is my language my layman's terms it basically means the same the same order remember we wrote down the order 2 by 3 3 by 2 you know 4 by 6 that was the order basically they have to have the same number of rows and columns and so that means that they're gonna have exactly the same order it's the same thing as the same size and shape but I'll just like the way that sounds better okay so let's look at a couple of examples of matrices that are that are equal or not let's say I had these two matrices 1 2 3 4 5 6 notice this matrix has 3 rows 1 2 3 and 2 columns 1 2 and I have another matrix okay and again I have 1 2 3 4 5 6 exactly the same numbers inside let me ask you this question are these two matrices equal to one another are they equal okay well you notice they have exactly the same numbers 1 2 3 4 5 6 they're all the same but they do not have the same size in the shape so these two things are not equal to each other they just aren't okay it doesn't matter if they match exactly element 4 element you can go across reading them if they don't look the same they don't live the same size and shape they're not equal okay they have to look exactly the same this guy has two rows and three columns obviously that's different than the three rows and two columns from before so those are not equal okay because they have different shape basically okay so let's look at another example let's say you had a matrix that was a square matrix and it was 31 inside and 79 inside 0 and 31 and I'm gonna write down another matrix and let's see if they're equal 31 79 10 and 31 and I asked you based on this definition above are these two matrices equal well 31 matches 79 matches zero does not match the corresponding element okay and 31 does match okay but and notice in this case that both of these are two by two matrices two rows two columns to rows to columns so they're both two by two they both have the same order and they both have the majority of the elements the same but they are not equal and the reason they're not equal is precisely because the zero does not equal the ten okay so you see what I'm saying here is that when you try to look and see if two matrices are equal to one another they have to look the same as far as the same shape they have to have the same number of rows in the same number of columns and the corresponding elements the corresponding elements here in every single place must be the same if they are not the same then they're not they're not equivalent that's just basically it okay now let's look at another one let's say I have 7 3 4 negative 1 2 3 okay that's a matrix let's put another matrix down with some different elements inside 7 3 4 negative 1 2 & 3 and I ask you are these two matrices equivalent well they have the same number of rows and columns to rows three columns to rows 3 columns 7 matches with 7 3 with 3 4 with 4 negative 1 with negative 1 2 with 2 3 with 3 because they have exactly the same size and shape because every single element is exactly equal to the corresponding element these two things are exactly the same they are equivalent okay and just to drill at home a little bit more 1 3 9 7 2 4 is it or is it not equal to 1 3 9 7 2 4 okay I hope you can see at this point that these guys are equal okay they have the same number of rows three rows three rows two columns two columns and everything is equivalent 3 with 3 9 with 9 7 & 7 2 with 2 4 with 4 okay if any of these numbers did not match the corresponding number just like in this case up here they would not be equal but because they have the same row number of rows and columns and because every single every single corresponding entry is the same they are equal so that's pretty important because a lot of times you'll be asked is this matrix equivalent to another one are they equal okay and you'll just have to go and look and make sure that there have all the same elements and everything else so you see there's there's not much math in this yet I mean it's it's not that hard there's some numbers in there we're just learning about what the matrix is later on we'll learn how to add them and subtract them and and and go about our business there okay and by the way just to give you a prelude adding and subtracting matrices are really really simple multiplication is a little more involved that's gonna throw you a little bit for a loop later adding and subtracting is a piece of cake you'll do that in your sleep okay so for the following problems here let's do two things just to get some practice let's give the order of the matrix okay and let's also we want to find a three sub two and we want to find a sub two sub three if we can a three two and A two three if possible let's find those and if it's not possible let's let's go ahead and look and see why it's not possible so for the first problem let's go ahead and just write the matrix down and the matrix that we're gonna be given here is going to be four and negative seven and five and negative six and eight and negative one that's my matrix so the first thing we want to do is find that work okay well we see right away that we have two rows and we also see that we have one two three columns and if you remember the order of a matrix is the rows with this X here meaning two by three rows comes first and then columns so the order is 2x3 okay order is 2x3 now let's try to go into the second part let's find a 3 2 and a 2 3 if possible when is a 3 2 a sub 3 sub 2 first number is the row and the second number is the column so we have to go to Row 3 Row 1 Row 2 there is no ro3 okay because there's no Row three this makes no sense at all so this does not exist or you can say it's not possible to find it or whatever it's because this matrix only has two rows so you can't find a sub 3 - because you're trying to find something in the third row there is no third row so you put that it doesn't exist now a sub 2 3 we're going to be able to find ok first number is the row second row second number is third column so row two third column here so the intersection is negative one okay a sub 2 sub 3 is going to be negative 1 so that's just what we're doing we're getting a little bit of practice with with doing these things identifying the order of the matrix getting comfortable looking at them and then pulling out you know an element here just to get a little bit more practice with it let's say our next matrix is a big matrix 1 negative 5 PI e 0 7 negative 6 negative PI negative 2 1/2 11 negative 1 yeah now part of my motivation for breaking the section out kind of by itself is just to get you comfortable looking at these matrices ok this is a collection of numbers don't be scared off by the fact that there's PI in here or negative PI I mean these are just numbers to 3.14159 on and on and on that's a number E is also a number that you remember back from algebra 2.71 la la la it goes on forever ok 1/2 is a number negative 1 fits the number they're just fractions so there's nothing weird here they're all numbers just because they have a couple of things PI and E I mean that's just depending on what your system of equations that you're trying to solve that we're going to learn how to do later is you may or may not have some goofy looking numbers in there but for the purpose of this example they're just it's just an abstract thing we're just looking at the matrix so how many rows do we have we have three rows one two three rows okay how many columns do we have one two three four columns so because we're trying to find the order of this matrix we we'll say that we have a 3x4 order because the rows always come first and the columns always come after that three by four okay now after we find the order let's go ahead and try to find a sub three two okay again the first number refers to the row you're on one two third row second number is the column whose first column here's second column third row second column just like a crossword puzzle one half okay and finally a sub two three okay first number is the row we're on the second row this is the column one two three second row third column negative six just like a crossword puzzle okay so these are the two elements and you could be asked on the testifying you know any element here and you'll know how to do it you basically just look at the intersection of the row and the column and you can pull out any element of a matrix and that's how you represent it with these little subscripts so when you first look at this though in a book especially when you flip open the page to matrices you'll see a bunch of square brackets and a bunch of stuff in there and then subscripts with I and J and maybe some numbers and it just looks confusing this is like a crossword puzzle that's all you're doing you're finding the intersection to pull out the element that you're interested in okay now in this set of problems what we want to do is we want to find the value for the variables and you'll see what I mean again in a second that make the matrix equation true ok so don't don't be worried about this ok it has a lot of fancy words the variables that make the matrix equation true it sounds it sounds crazy but let me show you what what we're doing and it's it's not gonna be a big deal at all if on a test and frequently you're given questions like this is why I'm doing them let's say I have 2x in here let's say I have negative 4 in here 6 negative 3y and let's say that they're claiming that this is equal to the matrix labeled by negative 10 negative 4 6 & 6 all right and the question is what values of x and y are needed to make this matrix equation true let's step back and look at what we have here this is a matrix equation a simple simple simple simple matrix equation okay it looks complicated but really all it means is you have a matrix here and you have a matrix here and we're claiming that they're equal to each other just like when you looked at it back in Algebra one we had the left side which had X's and Y's and exponents and division everything else we had a right side and we were claiming that they were equal and a lot of times you were just solving for the value of x and y that made them equal in your basic equations from Algebra one this is no different I have any matrix here and a matrix here and I'm claiming that they're equal I just need to figure out the values of x and y that make them equal okay now also earlier I told you most of the time matrices you're going to be dealing with numbers and not not variables and that is true that doesn't mean you cannot have variables in here I'm just telling you that most of the time they're matrices are just gonna have numbers inside and that's true but in this kind of problem usually they're there they're trying to throw you for a loop a little bit and trying to make you think about what what this means okay remember the fundamental thing about matrix matrices if they're equal the only way they can be equal to each other is if they have the same size and shape which they do two rows two columns two rows two columns so they have the same number of rows and columns and every single element must be the same as the corresponding element negative four is equal to negative four six is equal to six so you see the only way these two things can actually be equivalent matrices is if this element is equal to negative ten and if this element is equal to negative ten so all you have to write down is 2x is equal to negative ten and also negative 3y must be equal to 6 this element must be equal to this element this element must be equal to this element if that is true then that means that these two matrices really are equal to each other which we're claiming that they are so now you just solve for X I think you should know that you're just divide by 2 on both sides you get negative 5 and then here Y you're going to divide by negative 3 on both sides so you're gonna get negative 2 so if X is equal to 5 and Y is a its X is equal to negative 5 and Y is equal to negative 2 if those two things are true then this matrix is equal to this matrix if x and y have any different values than that those matrices are not going to be equal to each other so that's a very very simple matrix equation all it means is you have the left side and the right side and with matrices and they're equal to each other and you're just solving for the values that make them equal so the corresponding values the corresponding elements I should say have to be equivalent so you set up a little mini equations here and solve them for x and y it's a very common test problem let's do another one let's say we have a little bit bigger we have X plus 3 over here 2w minus 8y plus 1 4x plus 6z minus 3 and 3 Z this is a matrix okay with some elements inside I just happen to have more complicated elements there they're expressions in there and I'm claiming that this matrix is equal to this one over here 0 negative 6 negative 3 2 X 2 Z plus 4 and negative 21 ok so I have 2 matrices 2 2 matrices and I'm claiming they're equal they do have the same size and shape now remember this right here this is an element this thing here this is an element this is an element element element element even though they have pluses and minuses I mean this is an element here so I have three rows and two columns I have three rows and two columns so they do have the same size and shape and in order for them to be actually equal to each other that means every element must be equal to the corresponding element over here just like in the simpler examples we were before they all have to be equal so all you're going to do is you're going to write down some little equations and solve them for the values of x and y and z and w to make these two matrices equal to each other so let's do that you're going to see it's not a big deal x + 3 must be equal to 0 it has to be because that's the only way these two matrices can actually be equal to each other so to solve this thing you just say X is equal to negative 3 move the 3 over to the other side okay you've already solved for one of these of variables ok let's go to this one here 2w minus 8 has got to be equal to negative 6 because this is a corresponding element so what you'll have is 2 W is equal to moving the 8 over here so you add 8 getting rid of this over here you add 8 to the side negative 6 plus the 8 that you're adding over here is going to give you a positive 2 okay and then to solve for W you divide both sides by the 2 and you're gonna have 1 ok so W is going to be equal to 1 and you circle that is an answer so look you're already making progress over here y plus 1 is got to be equal to this element right here which is negative 3 okay and 2 to make it happen you move the one over here so you subtract one from both sides and you're gonna have negative 4 move the negative 1 over and I get a 3 minus 1 gives you negative 4 okay and looking over here you'll have Z minus 3 has got to be equal to the corresponding guy over here which is 2z plus 4 okay now let's see if these are equal let's go ahead and move this 3 over here so I have Z is equal to 2 Z add 3 over here 3 plus 4 gives me 7 ok let's move the 2z over here we're going to subtract it from both sides I'm going to have negative z is equal to 7 subtracting 2z over here subtracting 2z over here and then finally solving for z you're gonna have z is equal to negative 7 z is equal to negative 7 okay so I think we've actually found all of our variables because there was why Z and W we found X Y Z and W okay and we've equated this with this this with this this with this and this with this but you know we can continue we can continue doing these just to make sure that we check ourselves correctly let's say we wanted to do this one just to make sure it was right we would have 4x we've already solved for X we really don't have to do this but we could say that this element is equal to this element just to double-check ourselves so the way you're going to do it is you're gonna let's say move the four open to 4x over here so you have six is equal to subtracting 4x from both sides you'll have negative 2x 4x minus negative 4x on both sides we have 2x minus the 4x giving you negative 2x and then X is going to be equal to dividing by negative 2 you'll have negative 3 okay and notice the X is equal to negative 3 by that equation that's exactly what we found up here it makes sense there can only be one value of x that makes these things equal so you really don't have to do these but we're doing them to be to check ourselves really this elements the only one we haven't done it has to be equal to negative 21 so we have 3z & is equal to negative 21 and you just divide both sides z is equal to divide both sides by 3 you'll have negative 7 okay Z is equal to negative 7 that's exactly what we found here Z is equal to negative 7 so you know this is really a duplicate of what we've done here this is really a duplicate of what we've done here but we do them all just to make sure that we got the right answer okay and in this case basically all you're doing is finding the values of XY and Z and W that make each element equal across the board and then for these values these matrices really are equal if these met if these variables have any different values than these then the elements will not be the same and so these will be totally different matrices okay so in this section we've covered a very important topic we've introduced you to the concept of a matrix hopefully it's not as a scary of a thing is it maybe was a little while ago it's basically a collection of numbers you can have variables inside yes most of the time you'll see in the rest of the class that you're dealing with numbers that's all it's going to usually be in there and I've given you the punchline what we're going to do with these things is we're going to use in the solve systems of equations and you're gonna find out that they're not a big deal and it's actually easier to use to solve the systems of equations especially large systems than it is to do it by hand the other ways that you've learned and your basic algebra so it's a road map ahead we're going to learn how to add these things we're going to learn how to subtract them we're gonna learn how to multiply them we're gonna learn how to solve the equations and systems of equations and go on from there so it's a good introductory section watch it a couple times if you need to we're gonna baby step our way through it learn the operations learn the properties of the matrices and then of course we're gonna solve systems of equations with them which is really the main point at least in this course of using matrix algebra

Channel: Math and Science

Views: 209,271

Rating: 4.847856 out of 5

Keywords: matrices, matrix math, matrix algebra, add matrices, calculate, linear, matrix lesson, matrices lesson, algebra of matrices, matrix math calculator, matrix math problems, matrix math rules, matrix math example, matrix algebra rules, matrix algebra definition, add two matrices, what is a matrix, what are matrices, algebra, algebra 2, college algebra, linear algebra, calculate with matrices, matrix calculation

Id: 94mdM-OcjLg

Length: 41min 42sec (2502 seconds)

Published: Wed Feb 03 2016