Welcome back, everyone. So let's take a look at this example. Again, we've got these three system of equations, and we want to solve this, which means we want to find solutions for x, y, and z. The way we're going to do this is by writing a matrix in row echelon form. Remember, that's just that specific format where we have ones along the diagonal and zeros everywhere else. Let's go ahead and get started here. The first thing we have to do is convert this into a matrix just by pulling out all the numbers and coefficients. So this is going to be a 2, 4, 6, and then 24. This will become a 1, 5, 12, and then 60, and this will become a 3, 6, 15, and then 20. So remember, all we have to do is just work the equations from top to bottom, focus on one number at a time. Every time we get ones, we're going to try to focus on everything else, getting them to be zeros. So let's take a look at this first number here. I just want a one that's in this position. I've got a 2 there, so that's bad. So can I use any of my operations like swapping, multiplying, or adding to solve this? Actually, I can. So what you'll see here is that I can actually swap these two rows in order to get one in that position. So the first thing you can do is just swap. So we'll swap row 1 and row 2, and they'll basically just become each other. Alright? So, in other words, I'm just gonna rewrite this matrix, and this is gonna be 1, 5, 12, and then I've got 60. And this is gonna become 2, 4, 6, and then 24. And then I've got 3, 6, and then 15. And then finally, I've got 20. Alright? So I've gotten one of my numbers already. I've got 1. Now remember, rather than focusing on these other numbers and getting them to be ones, the next thing you wanna do is focus on getting everything underneath those ones to be zeros before you move on to the next one. So the way we do that, remember, is by adding. So what we're gonna do is, we're gonna take something to this row over here. And in order to cancel out this 2 to become a 0, I have to add it to something else. I can't add it to 3 because those things aren't gonna cancel out to 0. But what I can do is I can add row 2, and I can add it to a multiple of row 1 because I have this thing as 1 over here. So I can do this by multiplying row 1 by negative 2. And that will become my new row 2. Let's work this out real quick. So my row 2 is equal to, again, 2, 4, 6, and then 24. And then if I multiply negative 2 times row 1, that equals remember, I just multiply all these things by negative 2. So this is going to be negative 2, this would be negative 10, this will be negative 24, and this will be negative 120. Alright? Now, once you add those two things, what do you get? These things will cancel, and you'll just end up with 0, negative 6, and this would be negative 18, and this would be negative 96. Alright? So now that becomes my new row 2, and so you can just rewrite this matrix. So, really, what happens is I just get, 1, 5, 12, and 60, then the 3, 6, 15, and 20, because those things remain completely unchanged. And now the only thing that changes is row 2, which is now 0, negative 6, negative 18, and then negative 96. Alright? Notice how we got a 1, and now I've got one of my zeros. Now I'm just gonna focus on getting this thing to be 0, and I'm just gonna do a very similar step. Instead of adding something to row 2, I'm gonna add something to row 3. And so I'm gonna add row 3. And just like we added a multiple of row 1, we're gonna do the exact same thing. But instead of negative 2, all we're gonna do is we're gonna multiply or add to a multiple of negative 3 of row 1. Alright. So it's the same exact procedure, just trying to cancel out that 3 there. So what does this become? Well, row 3 again is just equal to I've got 3, 6, 15, 36, 15, and 20, and then negative 3 times row 1 equals I'm gonna have negative 3, This becomes negative 15, negative 36, and then negative 180.
Table of contents
- 0. Review of Algebra4h 16m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 6m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 19m
- 10. Combinatorics & Probability1h 45m
7. Systems of Equations & Matrices
Introduction to Matrices
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