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 2 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 going to show you in this video is that there are really just 3 of them that you need to know, and I'm going to 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 going to 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 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 going to rewrite this new row. Alright? So what happens is we're going to take all these numbers, multiply them by 2. This may end 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 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 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 going to do here is you're going to take this little r 2 You're going to take this little r 2, and I'm just going to add it to all the other numbers in little r one, and I'm just going to add these things straight down. And that now becomes my big R 2. So this should become 0, 10, and 30, and you're going to 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 sort of 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 going to take row 2, and we're going to swap it with row 3. So remember, this is just the notation for swapping. So all we're going to do here is this is going to be my r 2. This is going to be, sorry. This is going to be that's going to be r 2, and now it's just going to trade places with the 3rd row. Alright? So just going to rewrite this matrix over here. Remember, this the first row is going to 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 this row over here will go to the top. So this is going to be -1, 5, 9, and 3. Alright? So that is how you swap 2 rows. Alright? So now we're going to see here that this is our 3, and that was r 2 before. Let's take a look at the second one. So the second one says we're going to take r one. We're going to multiply it by a half, and then that's going to become the new r1. So this is really just multiplying something by a non-zero number. I actually want to point that out real quick here. You can only multiply by a non-zero number. You can't just multiply everything by 0 because it'd basically just be like almost deleting the equation. So you just can't do that. Alright? So let's take a look here. We're going to take this row 1, and then that's going to become the new r one. So what we can see here is that, remember, the rows the 2nd and third rows will remain completely unaffected. So those won't change at all. This would be 3, 8, -7, and 0, and this would be -1, 5, 9, and 3. So, that's what this b for b. And then what happens is for this row on the top, we're going to take all these numbers, and we're going to multiply them by half. So 2 becomes 1, the -6 would become -3, 4 become 2, and then the 10 would become 5. So this is what this new equation or matrix would look like. All right. So that is multiplication. The last thing we'll do is we'll do the addition. So we have that r two. We're going to add it to a multiple of r three, and that's going to become my big R 2. Alright? Let's take a look at this. So remember, what's going to happen here is the row 2 is going to get rewritten, which means that we can just rewrite the 1st and third rows. So in other words, we can just do the 2. By the way, we're not carrying over the changes that we made in a and b. This is going to be 2, -6, 4, and 10. And this is going to be, -1, 5, 9, and 3. So what's going on with this middle row over here? Well, remember, what this whole process is by adding one row to another, you're going to take all of these numbers here in row 2, and you're going to add them to 3 times all the numbers that are in row 3. Alright? So, actually, I should mention this is here r 3. So, basically, I'm going to take all these numbers here and multiply them by 3 and then add them to all these numbers over here. Alright? So this is how this is going to work out. We're going to come up with 4 numbers. Right? So this is going to the first one's going to be 3 +3 times -1. That's this pair over here. 2nd number is going to be 8 +3 times 5, just this row or this column. And the 3rd number is going to be -7 +3 times 9, which is going to be this pair. And then finally, you have 0 plus 3 times 3. Hopefully, you guys get the pattern. Right? So, basically, we're just going to go ahead and calculate all of these. And remember, this is just to do this operation over here. So this works out to 3 +- three, which turns out to just be 0. So this is going to be 0. This is going to be 8 +3 times 5, which is 15. That's going to be 23. That's the second number. The third one is going to be -7 plus 3 times 9, which is 27. That works out to positive 20. That's positive 20. And the last number is going to be 0 plus 3 times 3, which is just 9. Alright? So that's how to do these kinds of operations, where you are adding 1 row to another. Hopefully, that made sense. Thanks for watching, and I'll see you in the next one.