0:00
There you go.
00:01
This is a nice little fun systems of equations, but we're solving it with matrices.
00:07
So we have given matrix a, which is based off of the system right here.
00:12
Here's our system.
00:17
And we're told to write the solution, xy, as the product of two matrices, blank times blank, this is x, y.
00:27
And then of course find the solution so in order to find the solution let's let's go and do this so a know how i did this when i was in learning how to do matrices you take the identity matrix and you put it on the rights like this kind of augmented matrix here and then you basically using elementary operations elementary row operations you make this less left 2x2 matrix into the identity.
01:04
So how i'm going to do that, i'm going to get rid of this 5 here.
01:09
So i'm going to take the top equation and multiply it by negative 5 fourths and add it to the bottom equation here.
01:20
So that's going to give me 4, 2, because the top rows aren't changing at all.
01:27
This is going to end up being negative 5, so that goes to 0.
01:31
That's nice.
01:31
But then here we have to do some little work here.
01:33
This ends up being negative 10 fourths and then we're going to add it to three, this three, but that three, we want to add it to something that's divided to four parts.
01:47
So we'll make it 12 thirds or 12 fourths, pardon me.
01:52
And that's going to give us two fourths or one half.
01:55
So we get one half down here.
01:57
And then that means this is going to be, we're adding this one times negative five fourths to zero.
02:03
So we get negative fourths.
02:05
And then, of course, that time zero is zero to one.
02:09
So good.
02:10
We've done that.
02:11
And now we want to get rid of this two.
02:15
We want to basically diagonalize this matrix, make it a, which means get the leading, make for the leading row, that leading entry, zero ever else.
02:31
And same thing with this two.
02:34
Zero is all above this leading entry, the one half.
02:38
So to do that, we're going to take it and multiply it by a, i'll do this in a separate color times negative.
02:44
Let's say, i want to make that two.
02:45
So it's going to be four, right? yeah, that'll make it two.
02:48
So then we get four.
02:51
That's going to go to zero.
02:52
It's going to be zero, one half.
02:55
This is going to be negative five fourths and one, because that's not changing.
03:00
We're going to take this bottom row, multiply by negative four, add it to the top row.
03:05
So this is going to give us negative four times negative five fourths.
03:11
So that's going to be positive.
03:12
Five same color scheme positive five plus one that little one so that's going to give us six and we're going to do the same thing here that negative that one is multiplied by negative four at zero so we get negative four great now we want to take this and turn the left side into the identity matrix which means divide the top row by four and divide the bottom row by negative a half or multiply by two, or excuse me, divide by one half or multiplied by two...