Everyone, welcome back. So, we've spent a lot of time talking about systems of equations in the last few videos. We're going to shift gears a little bit. We're going to start talking about a new idea called a matrix, or the plural version is matrices. Now matrices can seem kind of scary at first because you'll see these big blobs of brackets with numbers inside them and all that stuff. But I'm actually going to break it down for you, and I'm going to show you that a matrix is really just a way to organize information or numbers into a grid-like pattern with rows and columns. Alright? So let's go ahead and get to it. I'm going to break it down for you. Alright? So we've got this matrix here, and the way we sort of define a matrix or we label the these parts of a matrix is by rows and columns. There are 2 rows in this matrix. We have 1 and 2. Those go horizontally, and we also have 3 columns in this matrix. Right? So 3 columns. So this is just 1 2 3. Alright? Now, so you might see this matrix referred to as a 2 by 3 matrix, 2 rows, 3 columns. Alright? Now, really, all this matrix is is just a way to sort of arrange numbers, and you'll actually see that the numbers are the exact same that they are in the system of equations over here. In this system of equations, we had 2 equations, and those things just became our 2 rows in our matrix. We also have three numbers in the columns, 2 for the coefficients and then one for the constants over here. Those three numbers really just became our 3 columns over here. So really, it's just a way to represent this information that's in the system just in a different way. Alright? And when you do this, when you have a system of equations represented as a matrix, it's called an augmented matrix. That's just a sort of a fancy word that you'll see in your textbooks and here in your classes. That's all that really means. Alright? So all we're really doing here is we're just sort of copying over the coefficients into this matrix, and we're sort of leaving out the variables. So we have negative 72. That becomes negative 72. Then we have this negative 3 and onetwo. That's negative 3 and onetwo. And then this negative 4 and 13 just becomes negative 4 and 13. So we're just copying over all the numbers, but we're just leaving out the coefficients. Because in a matrix, it's understood that these columns mean x and y coefficients. Now what you'll also see is you'll see this little black bar that's inside of these matrices, and this just means an equal sign. That's kind of what that means. So you'll see here that the really, a matrix is just a more compact way to represent all of the information that was in the system of equations. Alright? That's really all there is to it. Later on, we'll learn different ways to manipulate matrices and even do some operations with them. But for now, all we need to do is basically just turn a system into a matrix. Let's go ahead and take a look at an example problem here. So we're going to take this 2x plus 3y. So we're going to take this system of equations and represent this as a matrix. So we've got 2x plus negative 3y equals or, like, plus z equals negative 4. And then we got 6x plus 3y equals 13, and then y minus z equals 8. Alright? So it's really important that you actually lay out when you have a system of equations so that you have the coefficients and the variables that are on top of each other, x's with x's, y's with y's, and z's with z's. And so that's one of the things you may have to do before turning something into a matrix. Alright? But remember, really, all we're going to do is we're just sort of going to extract out the coefficients. So I'm going to have 3 columns here because I have 3 variables. Actually, I'm going to have 4 columns because I have 3 variables plus one constant on either side. So this is going to be like, the constants over here. Alright? So if you look at this first equation, what happens is I have a 2 in the x, negative 3 in the y, and then 1 in the z. So this is just going to be 2, negative 3, and then 1. The constant that goes on the other side is going to be negative 4. Alright? So that's for that first equation. Let's take a look at the second one. The second one has 6 in the x, 3 in the y, so 6x3y. But what about the z components? Here, we had one z over here. What happens if we just see nothing? Well, really, if you ever just see a blank space, it actually just means that there is, like, 0 z there. There's, like, a 0 there. So you don't just leave the matrix empty. You actually will just put a 0 there. Alright? So this will be 13. And then finally, what we're going to have is y minus z is 8. So here now what happens is that we have no x components. There's no x coefficient, so we just label that as 0. Then we have here is 1, and then minus z means negative 1. And over here, we have 8. So this is how you take this system of equations here and represent this as a matrix. So this would be the augmented matrix for this system of equations. Alright? That's it for this one, folks. Thanks for watching. I'll see you in the next one.