Welcome back, everyone. So back when we studied systems of equations, we saw lots of different ways to manipulate those equations. For example, we saw that we could swap the positions of 2 equations. We could also multiply equations by some number, like when we did the elimination method. We can also add equations together once we had those coefficients to be equal and opposite. Well, remember that a matrix is really just a representation of a system of equations. It's just two ways of writing the same information. So just as we did operations to these types of equations, we can also do operations on the rows of a matrix because remember, an equation is really just a row on a matrix. So we call these things row operations. What I'm gonna show you in this video is that there's really just three of them that you need to know, and I'm gonna break it down for you, showing you a bunch of examples. Alright? And there's also some new notation we'll learn as well. Let's get started with the first one, which is swapping 2 rows. This is by far the easiest operation, and it sounds exactly like what it sounds like. Right? We're just gonna be swapping the position of 2 rows just like we swap the position of 2 equations. So, here, all that happens is that we had -1, 2, and 9 on the top row, and now it just goes to the bottom. So this is -1, 2, and 9. And then the 2, 6, and 12 that was on the bottom, now it just goes to the top, 2, 6, and 12. Now you'll see some notation for this written in your textbooks with some little r's and big R's. Really, all this stuff says is little r means the old row, and big R means the new row once you're done doing that operation. And the subscripts just tell you which number or which row they're talking about. So, for example, this is little r one over here, and that little r one becomes big R two and vice versa. This little r two becomes big R one. So that's really all that's going on there. It's just showing you old versus new and the number of the row that it is. Alright? So that's the first one. Let's take a look at the second operation, which is multiplying one row by any non-zero number. Alright? So when we dealt with the system of equations in the elimination method, we could multiply an equation by some number. So, for example, we multiply this by 2. And what happens is we change all the coefficients. This would be -2x and this would be 4y, and this would be 18. Well, we can do the exact same thing to the rows or the numbers in the rows of a matrix. So really all this is we're going to take this little r two over here, and the notation for this is we're going to take little r two multiplied by 2, and that becomes now big R two. We're just gonna rewrite this new row. Alright? So what happens is we're gonna take all these numbers, multiply them by 2. This may ends up being -2, 4, and 18. Notice how all these numbers are the same because remember, this matrix is just representing the system of equations. Alright? Alright. So these numbers here are all the same. Alright. So, let's move on to now the last one, which is just adding some multiple of one row to another. Alright. So when we did this for systems of equations and we finally got their coefficients to be equal and opposite, we could add the equation straight down and cancel out or get rid of 1 of the variables. What we were left with is we were left with something like 10y equals 30. So now what we can do here is, with a matrix, we can do the same exact thing. Now when we did the first system, we always just delete or not even rewrite that first equation because we're only just worried about that one equation here. With the matrix, you can't just sort of delete a row, so you just rewrite it. Right? So the 2, 6, and 12, you just rewrite the 2, 6, and 12. Alright? But now what you're gonna do here is you're gonna take this little r 2. You're gonna take this little r 2, and I'm just gonna add it to all the other numbers in little r one, and I'm just gonna add these things straight down. And that now becomes my big R 2. So, this should become 0, 10, and 30, and you're gonna get exactly the same sort of numbers that you get on left and right. It's just another way to represent this system of equations. Now, unfortunately, this step here of adding, some kind of a non-zero multiple of one row to another is actually, unfortunately, the most common step, so it's good to get some good practice with this. I also want to mention one other thing here. The last two steps that we talked about are operations, the multiplying and adding. They only affect one row. It's the row that they're you're currently doing an operation on. The only time you're actually doing 2 rows or you're affecting 2 rows is when you're swapping them. So what you'll see here is that we rewrote, for example, the 2, 6, and 12. We changed we never it never changed the entire time. And that's because the only row that was changing was row 2. Alright? So, anyway, those are the 3 operations. Let's go ahead and get some more practice here using, this this, this sort of more a little bit more complicated matrix. So here we've got 2, -6, 4, 10. We've got this big matrix over here. Let's take a look at the first one. The first one says we're gonna take row 2, and we're gonna swap it with row 3. So remember, this is just the notation for swapping. So all we're gonna do here is this is gonna be my r 2. This is gonna be, sorry. This is gonna be that's gonna be r 2, and now it's just gonna trade places with the 3rd row. Alright? So just gonna rewrite this matrix over here. Remember, this the first row is gonna remain completely unaffected. So this is 2, -6, 4, 10. Just rewrite it. Now what happens is the 3 eights, -7, and 0, will actually go to the bottom. 3, 8, -7, and 0, and now what happens is the this row over here will go to the top. So this is gonna be -1, 5, 9, and 3.