An Introduction to Matrices - Matrix Algebra Tutor

Hi and welcome to the Matrix Algebra Tutor and in this class, we are going to focus on a part of Algebra that is sort of compartmentalized. A lot of times you will learn about this stuff in college Algebra. You will learn about it sometimes at the very end of an Algebra II class depending on your class and of course, when you are off doing engineering and other things, you will have a dedicated class in this called the linear Algebra. At least some of the topics in your linear Algebra class will be covered by everything on this DVD and additionally more things as well.

So the point is matrix Algebra which we are going to learn about over the next several hours is very integral and it is a part of Algebra that is very, very useful and it is a part of Algebra that can throw people for a loop, mainly because it looks completely different than anything you have 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 Xs and Ys and exponents and things like this but matrix Algebra or matrices, they usually do not 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 see these square brackets and you know that there are some sort of math going on here but it looks completely different than anything else you probably learned up until that point.

And so it looks sort of like gibberish and it looks like it is going to be difficult but the fact that the matter is once you take it one step at a time from the beginning and march through what these things are about and worry how to use them and how to manipulate them just like you had to learn how to manipulate your variables and your exponentials and everything else, we are going to learn about how to manipulate these matrix things.

And it is not going to be a big deal, that is the point, so this section of the class is the first section and what we are going to learn about mainly is what is a matrix. Let us write it down, let us talk about its components, and let us get comfortable looking at them. We are not going to be adding anything or subtracting anything. We are just going to learn about what a matrix is and why they are useful.

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? Let me ask you this, when you remember back to your Algebra from a long time ago, at some point you did study something called solving a system of equations, that is what you studied. It is covered in my DVDs. It is covered in every Algebra class that you are going to take whether it is college Algebra or Algebra II in high school or whatever, you will study something called solving a system of equations. That is a big fancy thing and all that it means is, if you remember back, you have of instead just one equation to solve like X+2=9 that would be one equation. A system of equations is usually more than one equation and 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 are trying to solve for two unknowns, okay X and Y let us say, or if you want to have a more complicated problem, you could have three variables you are trying to solve; X, Y, and Z, and because you have three variables, you must have three equations in order to solve for all the variables.

So the long and short of it is, you have to have as many equations to describe the system you are talking about as variables you are trying to solve, so as a practical example let us say in this room, this gas this air around my head right now, it has a certain temperature, it has a certain pressure, 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 us say.

So those are three variables, we could call them X, Y, and Z. If you are 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 us just say the humidity of the gas, so three variables 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 the system, and you would have to have three different equations to solve for those three variables if that is 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 learned something called addition, solving them by addition or sometimes it is called solving them by subtraction.

Basically that is just taking the equations and lining them up one under another and the long and short of it is, you can subtract them from one another and by doing that, you end up eliminating some variables and solving for some variables.

So what am I leading to why am I taking you down this memory lane of Algebra that you have studied 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, you already know a few techniques to solve those things, you have already done that before.

The main use of matrices or matrix Algebra is doing that solving a system of equations in a shorthand fast way, that is really what it is. So if you have to take one thing away from this section, at least from the beginning here, the entire point of this entire DVD course is not just to bore you with a bunch of matrices on the board, it is 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 in real life problems.

You can do it without all of that substitution, without all of that equation, addition, you are going to solve it using matrices and you are going to find out that once you figure out how to crank through it, it is going to be actually not too bad, and it is going to save your time because it is just going to 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 are writing all the equations down every time and solving them.

With the matrix, once we show you how, it is not going to be a big deal. So that is where we are going, that is the end game, the first few sections of this course are going to teach you what a matrix is, how to add them, subtract them, and everything else sort of leading up giving you the skills so once we get to the middle of the class here the DVD, you will learn how to actually solve systems of equations and understand what these things are for. Now matrices have use as far beyond that, but that is certainly the most important thing to you to remember now and certainly and probably the most practical reason that we use them and computers use matrices to solve these things all the time if you are going to write a computer program to solve the real problem for more than one equation, you are definitely going to use a matrix to solve it, no question about that.

So let us finally dive in the topic here and figure out what a matrix is okay. The big deal here is a matrix is nothing more than some rows and columns of numbers. For a minute even though I have kind of given you the end game of where we are going that we are 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 is where we are going to use these things for.

But for now, just open your mind and learn about what a matrix is and it simply a square bracket with some numbers inside. It is not that big of a deal, so let us write that down. A matrix contains rows and columns of numbers. Now, notice I did not say that it contains variables, I did not say that it contains exponents, I said it is rows and columns of numbers.

So when you think about that, even though they might look a little funny at first, we will get to that here in a second. It is really just some brackets filled with numbers and you have been dealing with numbers ever since you were a kid, so even though it looks foreign to you at first, it is not that bad, there is not a bunch of variables running around here for you to keep track of with the matrix. Usually you are going to be dealing with numbers and numbers are much easier to wrap the brain around, usually and than a foreign concept with variables.

So what is a matrix? The easiest way to do it is to write one down. Here is what a simple matrix would look like. You put a square bracket and there are some numbers inside. Let us put one, two, three, and four, congratulations, we have just written your first matrix down. Now the first thing most people want to do when they first learn about matrices is; you will see these numbers here and it is confusing, there is no addition, subtraction, division, there is no operation in here, so it looks a little bit odd, there is no equal sign anywhere okay and that is a bunch of numbers but it just does not look like that it is very useful.

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 it for what it is really intended just like you had to learn what a negative number was and that was a really complicated thing way back in the day. You are going to have to learn what these things look like.

So, these things are called matrices and notice that there are some rows, there are two rows in this matrix and there are two columns in this matrix. So you will see here that you have two rows in this matrix and you have here two columns. Two rows and two columns, why is that important? Because you will 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 were writing these things down, you refer to the order of the matrices or the order of the matrix just to the describe how big it is, what its shape is.

So what you would say in this case is that the order of this matrix which you will probably have to do in your test, you will be given a matrix and you will say what is the order of this matrix? and you will be like what is that? You have to learn the definition. The order basically just says how many rows? How many columns?.

The first number is the rows, and the second number is the columns, okay so this is two by two, the X is not a variable X. It is like a two by four piece of lumber or something, that is what the X means here, so it is a two by two matrix, two rows and two columns. So you see, this is not rocket science. There is no variables here, there is nothing complicated here, there are 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 have already given you the punch line. You see this matrix, later on we are going to learn, this matrix can be use to represent an equation.

Actually it can be use to represent a system of equations, just like you were solving those systems of equations when you first learned Algebra. We are going to use these matrices to solve them easier, to basically solve them easier, and that is really what it is all about. So this is a two by two matrix, and it is called the order of the matrix if you have a question on your test what is the order of the matrix? You will write down how many rows, how many columns, put an X in the middle and you read it two by two or three by two or five by four or whatever, that is the order of the matrix. So, let us go ahead and write down what the order of a few other matrices would be just to get a little bit of practice.

Let us say we have a matrix, that looks like this; zero, negative one, three, zero, one, and four. Notice I do have a negative number in here, and notice I do have some zeroes in here, so I said the matrix is usually going to be full of numbers. These numbers can be negative-positive, they can be decimals, they can be fractions, they can be any number that you want really; any real number, that is what we are going to use this things for.

The zeroes in here, we still have to put them in their place, and again this would represent, I am giving you the punch line a little bit, later on we will see that this represents a system of equations that you would then use this matrix to solve. So that is where we are going with this. If you were trying to find the order of this matrix, all you got to do is look and say look it has two rows by three columns, one, two, three columns, so it is a two by three order; two by three, so that is another matrix.

Just to give you a little bit more practice, let us go ahead and write another one. Let us switch colors a little bit. What if I have; notice I am kind of drawing a vertical skyscraper matrix here, one, three, let us see 74 just for kicks, 29 let us say negative five and negative one. So we got some big numbers, we got some negative numbers here, but again it is the same thing, what is the order of this matrix, if you were asked that on your test, you will say how many rows do I have? I have three rows so that number goes first and two columns, so it is a three by two order.

So we have not really done anything with these things yet, we just talked about the fact that we have this thing called an order which is the number of rows and the number of columns, and that is sort of just the definition more than anything else. We have not done any operations with these things. We are just writing them down. You have no idea at this point how you would use this thing to solve a system of equations but that is fine because we will get there.

Now let us continue learning about the matrix, and learn about the elements of the matrix. 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. So this thing, this entire entity is called the matrix and everything inside of it, every single number is called an element of the matrix so if you are 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 are talking about, you know that you are going to be looking inside of this thing trying to pull a number out, so that is what you are going to be doing. So the elements of a matrix are the numbers inside. So let us go ahead and draw that. The easiest way to do this is by example. Let us say you have, let me start 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 the subscript J.

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 and what all this stuff mean. This is just used to identify what element of the matrix you are talking about, and this is very, very simple. Let me show you what I am talking about here by example. Let us say we have a matrix, one, two, three, four, five, six, seven, eight, and nine; before we go any further, what would the order of this matrix be? Just think about that, you got to look at the rows and the columns, there are three rows and there are three columns so it is a three by three; 3x3, it is a 3x3 order matrix.

Now, the elements of the matrix are identified by A, usually the books use A with the subscript I and a subscript J. Do not get worried about this okay. All it means is this number, the first one, the first subscript identifies the row that you are talking about and this number identifies the column that you are talking about and once you have identified the row and the column that you are talking about, you can identify any number uniquely in here.

So all you are doing is; to pull out an element or to identify an element of a matrix, you are going to generally have to represent some variable really just to show that you are talking about an element of the matrix and you are going to have two little subscripts. The first subscript is going to identify the row you are talking about, is it this row? Is it this row? Or is it this row? The second number is going to talk about the column that you are referring to and the intersection of those two things is the element that you are talking about.

So to put it in more concrete terms, let us practice. Let us say I am trying to find element A sub one; one, on a test if you are given this matrix and you were told or asked what is the element denoted by A one, one? So all you have to do is realize the first number is the row, so it is the first row. This is row one, row, two , and row three, this is column one, column two, and column three, so you are in the row number one, up here.

Now in order to find out what element you mean which is what number you need, you are in column number one, so this is column number one, this is row number one, so this guy right here is the element that you are 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 is the element you are talking about. Because frequently when you are dealing with the matrix, you might have to refer to the elements of the matrix. You might have a need to write an equation involving the elements of the matrix so this little subscript notation is how you do that. It is like a crossword puzzle, I mean really it is no different than a crossword puzzle. Three across, two down, I mean it is really all it is, row and column, that is how you identify the elements of that thing.

So, what if you were trying to find A one, three. First number is the row, so I am on the row number one, second number is the column; column number three, so the element is three. Let us say I had to find A two, two, and by the way when I say A here, A here, and A here, that is if I am calling this matrix, matrix A. I could be calling it that and labeling it matrix A and these little subscripts denote the elements of the matrix A. That is why if you are going to call this matrix, matrix B or matrix Z or matrix G then you would have G with the subscripts, one-two, one-three, or whatever depending on what you actually labeled the thing, we are just using A because it is easy.

So row two column two, here is row two, here is column two and intersection is five. A two, three, row two column three, here is row number two, here is column number three, that is a six, and then let us say A three, one, and it is not really the number you care about, it is the individual digits here, so row three over here and column number one, intersection is seven. Okay, and finally, let us say I am going to find A three, two, so I look, the first digit is the row, the second digit is the column, so the first digit is row three, so I am going to row three, column number two, this is column number two, the intersection is eight, so the answer eight.

So we can come up with one, three, five, six, seven, and eight, and basically you can do this for any element of the matrix. If the matrix had let us say this matrix was huge, this matrix could have a hundred rows and 519 columns, I do not know why you would ever need to create a matrix that big but you could if you wanted to and so in order to look at the individual elements of that matrix the individual numbers, all you need to know is two numbers, the row it is on and the column it is on. Once you have that, you are in good shape, so this is a good introduction to matrices here and the rest of the section, we are going to talk a little bit more about when matrices are equal, when the different matrices are equal, and just get a little bit more practice with identifying the elements of the matrix here.

Now something else is 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 are called square matrix, I do not even think that I have mentioned that before but that is something that you would probably need to remember, when it has the same number of rows and columns, it is going to like a square, that is called the square matrix.

If it has different number of rows and columns which can always be the case and it is a not square matrix, it 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 asked 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 have 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. So let us write that down on, 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, that to me is everyday language, and everybody is going to understand. Basically, what this means and you will see there are a lot in your book, I will just go ahead and put a dotted line around it, same size and shape. This is my language, my layman terms, it basically means the same order.

Remember we wrote down the order; 2x3, 3x2, 4x6 that was the order. Basically they have to have the same number of rows and columns and so that means that they are going to have exactly the same order, it is the same thing as the same size and shape and I will just like the way that sounds better. So, let us look at a couple of examples of matrixes that are equal or not. Let us say I had these two matrices; one, two, three, four, five, six, and notice this matrix has three rows, one, two, three, and two columns, one, two. And I have another matrix, and again, I have one, two, three, four, five, six, exactly the same numbers inside. Let me ask you this question, are these two matrices equal to one another, are they equal?

You notice they have exactly the same numbers; one, two, three, four, five, six, they are all the same but they do not have the same size and shape so these two things are not equal to each other, they just are not. Okay, it does not matter if they match exactly element for element, you can go across reading them, if they do not look the same, they do not have the same size and shape, and they are not equal. They have to look exactly the same.

This guy has two rows and three columns, obviously that is different than the three rows and two columns from before, so those are not equal. Because they have different shape basically. So let us look at another example; let us say you had a matrix that was a square matrix and that was 31 inside and 79 inside, zero and 31, and I am going to write down another matrix and let us see if they are equal. 31, 79, 10 and 31, I ask you, based on this definition above are these two matrices equal? 31 matches, 79 matches, zero does not match the corresponding element and 31 does match. But notice in this case that both of these are two by two matrices, two rows, two columns, two rows, two columns, so they are 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 are not equal is precisely because the zero does not equal to 10.

So you see what I am 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 shape. They have to have the same numbers of rows and the same number of columns and the corresponding elements here in every single place must be the same. If they are not the same then they are not equivalent, that is just basically it. Now let us look at another one. Let us say I have; seven, three, four, negative one, two, three, that is a matrix and let us put another matrix down with some different elements inside; seven, three, four, negative, one, two, and three. I ask you, are these two matrices equivalent? They have the same number of rows and columns, two rows, three columns, two rows, three columns, seven matches with seven, three with three, four with four, negative one with negative one, two with two, three with three.

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, and just to drill it home a little bit more, one, three, nine, seven, two, four, is it or is it not equal to one, three, nine, seven, two, four. I hope you can see at this point that these guys are equal. They have the same number of rows, three rows, three rows, two columns, two columns, and everything is equivalent; three with three, nine with nine, seven and seven, two with two, four with four.

If anyone 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 number of rows and columns and because every single corresponding entry is the same, they are equal, so that is pretty important because a lot of times you will be asked, is this matrix equivalent to another one? Are they equal? You will just have to go and look and make sure that they have all the same elements and everything else. So you see there is not much math in this yet. I mean it is not that hard, there are some numbers in there, we were just learning about what the matrix is. Later on, we will learn how to add them and subtract them and go about our business there. By the way just to give you a prelude, adding and subtracting matrices is really, really simple.

Multiplications are little more involved, that is going throw you a little bit for a loop later. Adding and subtracting is a piece of cake, you can do that in your sleep. So for the following problems here, let us do two things, just to get some practice. Let us give the order of the matrix. And let us also; we wanted 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 us find those and if it is not possible let us go ahead and look and see why it is not possible.

So for the first problem, let us go ahead and just write the matrix down and the matrix that we are going to be given here is going to be four and negative seven and five and negative six and eight and negative one. That is my matrix, so the first thing we want to do is find the order. 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 the matrix is the rows with this X here meaning 2x3.

Rows come first, then columns, so the order is 2x3. Now let us try to go into the second part. Let us find A three, two and A two, three if possible. What is A three, two; A sub three sub two? First number is the row and the second number is the column so we have to go to row three; row one, row, two. There is no row three. Because there is no row three, this makes no sense at all. So this does not exist or you can say it is not possible to find it or whatever. It is because this matrix only has two rows, so you cannot find A sub three two because you are trying to find something in the third row, there is not third row.

So you put that it does not exist. Now A sub two three, we are going to be able to find. 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 two, sub three is going to be negative one. So that is just what we are doing we were getting a little bit of practice with doing these things, identifying the order of the matrix, getting comfortable looking at them and then pulling out an element here just to get a little bit more practice with it.

Let us say our next matrix is a big matrix; one, negative five, pi, E, zero, seven, negative six, negative pi, negative two, one half, eleven, negative one fifth. Now part of my motivation for breaking this section out kind of by itself is just to get you comfortable looking at these matrices. This is a collection of numbers, do not be scared off by the fact that there is pi in here or negative pi. I mean these are just numbers too; 3.14159, on and on and one, that is a number. E is also a number, that you remember back from Algebra, 2.71, it goes on forever.

One half is a number, negative one fifth is a number, they are just fractions, so there is nothing weird here. They are all numbers. Just because they have a couple of things pi and E, that is just depending on what your systems of equations that you are trying to solve that we are 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, that is just an abstract thing. We are just looking in the matrix. So, how many rows do we have? We have three rows; one, two, three rows. How many columns do we have? One, two, three, four columns, so because we are trying to find the order of this matrix, we will say that we have a 3x4 order because the rows always come first and the columns always come after that; 3x4.

Now after we find the order, let us go ahead and try to find A sub three, two. Again the first number refers to the row you are on; one, two, third row, second number is the column. Here is the first column, and here is the second column. Third row, second column; just like a crossword puzzle, one half. And finally A sub two, three. First number is the row, we are on the second row. This is the column; one, two, three, second row, third column, negative six. Just like a crossword puzzle so these are the two elements and you could be asked then to testify any element here and you will 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 the matrix and that is how you represent it with these little subscripts.

So when you first look at this though, in a book especially when you flip over in the page to matrices, you will 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, it is all you were doing. You are finding the intersection to pull out the element that you are interested in.

Now, in this set of problems, what we want to do is we want to find the value for the variables and you will see what I mean again in a second. That makes the matrix equation true. So do not be worried about this. It has a lot of fancy words, the variables that make the matrix equation true, it sounds crazy but let me show you what we are doing and it is not going to be big deal at all. If on a test, and frequently you are given questions like this is why doing them. Let us say I have two X in here, let us say you have a negative four in here, six, negative three Y, and let us say that they are claiming that this is equal to the matrix labeled by negative 10, negative four, six, and six, all right and the question is what values of X and Y are needed to make this matrix equation true? Let us step back and look at what we have here.

This is a matrix equation, a simple, simple, simple, simple matrix equation. It looks complicated but really all it means is you have a matrix here and you have a matrix here and we are claiming that they are equal to each other just like when you looked at equations back in Algebra I, we had the left side which had Xs and Ys and exponents and division everything else. We had a right side and we were claiming that they were equal. Alot of times you were just solving for the value of X and Y that made them equal in your basic equations from Algebra I. This is no different. I have a matrix here and a matrix here and I am claiming that they are equal. I just need to figure out the values of X and Y that make them equal.

Now also earlier, I told you most of the time, matrices are going to be dealing with numbers and not variables and that is true. That does not mean you cannot have variables in here. I am just telling you that most of the time your matrices are just going to have numbers inside and that is true, but in this kind of problem, usually they are trying to throw you for a loop a little bit and trying to make you think about what this means.

Remember the fundamental thing about matrices. If they are equal, the only way that 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 10 and if this element is equal to negative 10, so all you have to write down is two X is equal to negative 10, and also negative three Y must be equal to six. This element must be equal to this element, and 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 are claiming that they are, so now you just solve for X. I think you should know, that you just divide by two on both sides, you get negative five and then here Y, you are going to divide by negative three on both sides, so you are going to get negative two. So if X is equal to negative five and Y is equal to negative two, 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 is a very, very simple matrix equation, all it means is you have a left side and a right side with matrices and they are equal to each other and you are just solving for the values that make them equal so the corresponding elements have to be equivalent.

So you set up a little mini equations here and solve them for X and Y, so very common test problem. Let us do another one. Let us say we have a little bit bigger, we have X plus three over here, two W minus eight, Y plus one, four X plus six, Z minus three, and three Z. This is a matrix okay with some elements inside. I just happen to have more complicated elements, there are expressions in there. And I am claiming that this matrix is equal to this one over here; zero, negative six, negative three, two X, two Z plus four, and negative 21. So I have two matrices and I am claiming that they are equal, they do the 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 that were doing before. They all have to be equal, so all you are going do id you are 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 us do that, you are going to see that it is not a big deal. X plus three must be equal to zero, it has to be because that is 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 three, move the three over to the other side. You have already solved for one of the variables. Let us go to this one here; two W minus eight has got to be equal to negative six because this is a corresponding element, so what you will have is two W is equal to moving the eight over here, so you add eight getting rid of this over here, you add eight to this side negative six plus the eight that you are adding over here is going to give you a positive two. To solve for W, you divide both sides by the two and you are going to have one.

Okay so W is going to be equal to one and you circle that is an answer. So look, you are already making progress. Over here, Y plus one has got to be equal to this element right here which is negative three. To make it happen, you move the one over here, so you subtract the one from both sides and you are going to have negative four, move the negative one over and negative three minus one gives you a negative four.

And looking over here, you will have Z minus three has got to be equal to the corresponding guy over here which is two Z plus four. Now let us see if these are equal. Let us go ahead and move this three over here, so I have Z is equal to two Z, add a three over here, there plus four gives me seven. Let us move the two Z over here. We are going to subtract it from both sides and I am going to have negative Z is equal to seven, subtracting two Z over here, subtracting two Z over here, and then finally solving for Z, you are going to have Z is equal to negative seven.

So I think we have actually found all of our variables because there is X, Y, Z, and W. We found X, Y, Z, and W. And we have equated this with this, and this with this, and this with this, and this with this, but we could continue doing these just to make sure that we check ourselves correctly. Let us say we wanted to do this one just to make sure it was right, we would have four X, we have already solved for X and we really do not 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 are going to do it , is your going to let us say move the four over the four X over here, so you are going to have six is equal to-- Subtracting four X from both sides, you will have negative two X, for X minus negative four X on both sides so you have two X minus the four X giving you a negative two X and then X is going to be equal to--divided by negative two, you will have negative three.

And notice that X is equal to negative three. By that equation and that is exactly what we found up here. It makes sense. There can only be one value of X that makes this things equal, so you really do not have to do these, but we are doing them to check ourselves really.

This element is the only one that we have not done and it has to be equal to negative 21, so we have three Z is equal to negative 21 and you just divide both sides, Z is equal to divide both sides by three, you will have a negative seven. Z is equal to negative seven, that is exactly what we found here, Z is equal to negative seven so this is really a duplicate of what we have done here but we do them all just to make sure that we got the right answers, and in this case, basically, all you are doing is finding the values of X, Y, and Z and W that make each element equal across the board and then for these values, these matrices really are equal, if these variables have any different values than this then the elements would not be the same so these will be totally different matrices.

So in this section, we have covered a very important topic. We have introduce you to the concept of the matrix. Hopefully, it is not as scary of a thing as it may be was a little while ago. It is basically a collection of numbers. You can have variables inside, yes. Most of the time, you will see in the rest of the class, that you are dealing with numbers, that is all it is going to usually be in there and I have given you the punch line. What we are going to do with these things is we were going to use them to solve systems of equations and you are going to find out that they are not a big deal and it is actually easier to use them to solve the systems of equations especially large systems than it is to do it by hand the other ways that you have learned in your basic Algebra.

So as a road map ahead, we are going to learn how to add these things, we are going to learn how to subtract them, we are going to learn how to multiply them, we are going to learn how to solve equations and systems of equations and go on from there, so it is a good introductory section. Watch it a couple of times if you need to. We are going to baby step our way through it; learn the operations, learn the properties of the matrices and then of course, we are going to solve systems of equations with them which is really the main point, at least in this course of using matrix Algebra.

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