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Perhaps even more interesting than finding the inverse of a

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matrix is trying to determine when an inverse of a matrix

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doesn't exist. Or when it's undefined.

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And a square matrix for which there is no inverse, of which

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an inverse is undefined is called a singular matrix.

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So let's think about what a singular matrix will look

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like, and how that applies to the different problems that

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we've address using matrices.

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So if I had the other 2 by 2, because that's

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just a simpler example.

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But it carries over into really any size square matrix.

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So let's take our 2 by 2 matrix.

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And the elements are a, b, c and d.

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What's the inverse of that matrix?

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This hopefully is a bit of second nature to you now.

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It's 1 over the determinant of a, times the adjoint of a.

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And in this case, you just switch these two terms. So you

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have a d and an a.

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And you make these two terms negative.

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So you have minus c and minus b.

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So my question to you is, what will make this entire

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expression undefined?

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Well it doesn't matter what numbers I have. If I have

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numbers here that make a defined, then I can obviously

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swap them or make them negative, and it won't change

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this part of the expression.

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But what would create a problem is if we attempted to

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divide by 0 here.

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If the determinant of the matrix A were undefined.

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So A inverse is undefined, if and only if-- and in math they

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sometimes write it if with two f's-- if and only if the

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determinant of A is equal to 0.

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So the other way to view that is, if a determinant of any

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matrix is equal to 0, then that matrix is a singular

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matrix, and it has no inverse, or the inverse is undefined.

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So let's think about in conceptual terms, at least the

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two problems that we've looked at, what a 0 determinant

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means, and see if we can get a little bit of intuition for

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why there is no inverse.

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So what is a 0 determinant?

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In this case, what's a determinant of this 2 by 2?

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Well the determinant of A is equal to what?

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It's equal to ad minus bc.

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So this matrix is singular, or it has no inverse, if this

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expression is equal to 0.

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So let me write that over here.

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So if ad is equal to bc-- or we can just manipulate things,

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and we could say if a/b is equal to c/d-- I just divided

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both sides by b, and divided both sides by d-- so if the

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ratio of a:b is the same as the ratio of c:d, then this

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will have no inverse.

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Or another way we could write this expression, if a/c-- if I

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divide both sides by c, and divide both sides by d-- is

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equal to b/d.

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So another way that this would be singular is if-- and it's

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actually the same way.

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If this is true, then this is true.

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These are the same.

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Just a little bit of algebraic manipulation.

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But if the ratio of a:c is equal to the ratio of b:d, and

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you can think about why that's the same thing.

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The ratio of a:b being the same thing as

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the ratio of c:d.

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But anyway, I don't want to confuse you.

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But let's think about how that translates into some of the

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problems that we looked at.

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So let's say that we wanted to look at the problem-- Let's

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say that we had this matrix representing the linear

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equation problem.

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Well, actually, this would be either one.

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So I have a, b, c, d times x, y Is equal to two other

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numbers that we haven't used yet, e and f.

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So if we have this matrix equation representing the

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linear equation problem, then the linear equation problem

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would be translated a times x plus b times y is equal to e.

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And c times x plus d times y is equal to f.

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And we would want to see where these two intersect.

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That would be the solution, the vector

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solution to this equation.

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And so, just to get a visual understanding of what these

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two lines look like, let's put it into the

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slope y-intercept form.

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So this would become what?

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In this case, y is equal to what?

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y is equal to minus a/b, x plus e/b.

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I'm just skipping some steps.

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But you subtract ax from both sides.

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And then divide both sides by b, and you get that.

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And then this equation, if you put it in the same form, just

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solve for y.

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You get y is equal to minus c/d x plus f/y.

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So let's think about this.

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I should probably change colors because it looks too--

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Let's think about what these two equations would look like

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if this holds.

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And we said if this holds, then we have no determinant,

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and this becomes a singular matrix, and it has no inverse.

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And since it has no inverse, you can't solve this equation

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by multiplying both sides by the inverse, because the

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inverse doesn't exist.

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So let's think about this.

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If this is true, we have no determinant, but what does

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that mean intuitively in terms of these equations?

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Well if a/b is equal to c/d, these two lines will have the

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same slope.

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They'll have the same slope.

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So if these two expressions are different, then what do we

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know about them?

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If two lines that have the same slope and different

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y-intercepts, they're parallel to each other, and they will

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never, ever intersect.

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So let me draw that, just so you get the-- this top line--

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They don't have to be positive numbers, but since this has a

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negative, I'll draw it as a negative slope.

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So that's the first line.

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And its y-intercept will be e/b.

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That's this line right here.

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And then the second line-- let me do it in another color-- I

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don't know if it's going to be above or below that line, but

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it's going to be parallel.

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It'll look something like this.

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And that line's y-intercept-- so that's this line-- that

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line's y intercept is going to be f/y.

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So if e/b and f/y are different terms, but both

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lines have the same equation, they're going to be parallel

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and they'll never intersect.

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So there actually would be no solution.

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If someone told you-- just the traditional way that you've

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done it, either through substitution, or through

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adding or subtracting the linear equations-- you

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wouldn't be able to find a solution where these two

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intersect, if a/b is equal to c/d.

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So one way to view the singular matrix is that you

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have parallel lines.

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Well then you might say, hey Sal, but these two lines would

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intersect if e/b equaled f/y.

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If this and this were the same, then these would

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actually be the identical lines.

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And not only would they intersect, they would

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intersect in an infinite number of places.

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But still you would have no unique solution.

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You'd have no one solution to this equation.

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It would be true at all values of x and y.

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So you can kind of view it when you apply the matrices to

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this problem.

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The matrix is singular, if the two lines that are being

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represented are either parallel, or they are the

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exact same line.

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They're parallel and not intersecting at all.

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Or they are the exact same line, and they intersect at an

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infinite number of points.

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And so it kind of makes sense that the A

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inverse wasn't defined.

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So let's think about this in the context of the linear

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combinations of vectors.

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That's not what I wanted to use to erase it.

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So when we think of this problem in terms of linear

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combination of factors, we can think of it like this.

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That this is the same thing as the vector ac times x plus the

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00:09:15,009 --> 00:09:25,500
vector bd times y, is equal to the vector ef.

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So let's think about it a little bit.

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We're saying, is there some combination of the vector ac

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and the vector bd that equals the vector ef.

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But we just said that if we have no inverse here, we know

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that because the determinant is 0.

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And if the determinant is 0, then we know in this situation

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that a/c must equal b/d.

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So a/c is equal to b/d.

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So what does that tell us?

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Well let me draw it.

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And maybe numbers would be more helpful here.

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But I think you'll get the intuition.

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I'll just draw the first quadrant.

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I'll just assume both sectors are in the first quadrant.

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Let me draw.

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00:10:10,835 --> 00:10:17,990


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00:10:17,990 --> 00:10:19,590
The vector ac.

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00:10:19,590 --> 00:10:20,754
Let's say that this is a.

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00:10:20,754 --> 00:10:23,072
Let me do it in a different color.

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00:10:23,072 --> 00:10:24,860
So I'm gonna draw the vector ac.

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00:10:24,860 --> 00:10:31,539
So if this is a, and this is c, then the vector

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00:10:31,539 --> 00:10:33,870
ac looks like that.

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00:10:33,870 --> 00:10:34,259
Let me draw it.

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00:10:34,259 --> 00:10:36,210
I want to make this neat.

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00:10:36,210 --> 00:10:40,440
The vector ac is like that.

192
00:10:40,440 --> 00:10:43,120
And then we have the arrow.

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00:10:43,120 --> 00:10:44,950
And what would the vector bd look like?

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00:10:44,950 --> 00:10:49,700


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00:10:49,700 --> 00:10:54,310
Well the vector bd-- And I could draw

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00:10:54,309 --> 00:10:55,209
it arbitrarily someplace.

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00:10:55,210 --> 00:10:58,580
But we're assuming that there's no derivative-- sorry,

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00:10:58,580 --> 00:10:59,660
no determinant.

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00:10:59,659 --> 00:11:01,329
Have I been saying derivative the whole time?

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00:11:01,330 --> 00:11:02,090
I hope not.

201
00:11:02,090 --> 00:11:03,379
Well, we're assuming that there's no

202
00:11:03,379 --> 00:11:06,070
determinant to this matrix.

203
00:11:06,070 --> 00:11:08,010
So if there's no determinant, we know that

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00:11:08,009 --> 00:11:11,509
a/c is equal to b/d.

205
00:11:11,509 --> 00:11:16,120
Or another way to view it is that c/d is equal to d/b.

206
00:11:16,120 --> 00:11:17,929
But what that tells you is that both of these vectors

207
00:11:17,929 --> 00:11:19,349
kind of have the same slope.

208
00:11:19,350 --> 00:11:22,580
So if they both start at point 0, they're going to go in the

209
00:11:22,580 --> 00:11:23,139
same direction.

210
00:11:23,139 --> 00:11:25,500
They might have a different magnitude, but they're going

211
00:11:25,500 --> 00:11:27,080
to go in the same direction.

212
00:11:27,080 --> 00:11:37,009
So if this is point b, and this is point d, vector bd is

213
00:11:37,009 --> 00:11:39,539
going to be here.

214
00:11:39,539 --> 00:11:42,049
And if that's not obvious to you, think a little bit about

215
00:11:42,049 --> 00:11:46,079
why these two vectors, if this is true, are going to point in

216
00:11:46,080 --> 00:11:48,200
the same direction.

217
00:11:48,200 --> 00:11:52,020
So that vector is going to essentially overlap.

218
00:11:52,019 --> 00:11:56,039
It's going to have the same direction as this vector, but

219
00:11:56,039 --> 00:11:59,259
it's just going to have a different magnitude.

220
00:11:59,259 --> 00:12:00,730
It might have the same magnitude.

221
00:12:00,730 --> 00:12:04,480
So my question to you is, vector ef, we don't know where

222
00:12:04,480 --> 00:12:05,600
vector ef is.

223
00:12:05,600 --> 00:12:08,190
Well let's pick some arbitrary point.

224
00:12:08,190 --> 00:12:12,080
Let's say that this is e, and this is f.

225
00:12:12,080 --> 00:12:13,700
So this is vector ef up there.

226
00:12:13,700 --> 00:12:17,129
Let me do it in a different color.

227
00:12:17,129 --> 00:12:19,250
Vector ef, let's say it's there.

228
00:12:19,250 --> 00:12:23,419


229
00:12:23,419 --> 00:12:26,509
So my question to you is, if these two vectors are in the

230
00:12:26,509 --> 00:12:27,309
same direction.

231
00:12:27,309 --> 00:12:28,849
Maybe of different magnitude.

232
00:12:28,850 --> 00:12:32,620
Is there any way that you can add or subtract combinations

233
00:12:32,620 --> 00:12:35,139
of these two vectors to get to this vector?

234
00:12:35,139 --> 00:12:37,419
Well no, you can scale these vectors and add them.

235
00:12:37,419 --> 00:12:39,649
And all you're going to do is kind of move along this line.

236
00:12:39,649 --> 00:12:41,610
You can get to any other vector.

237
00:12:41,610 --> 00:12:43,800
There's a multiple of one of these vectors.

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00:12:43,799 --> 00:12:46,990
But because these are the exact same direction, you

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00:12:46,990 --> 00:12:49,560
can't get to any vector that's in a different direction.

240
00:12:49,559 --> 00:12:53,049
So if this vector is in a different direction, there's

241
00:12:53,049 --> 00:12:54,259
no solution here.

242
00:12:54,259 --> 00:13:00,529
If this vector just happened to be in the same direction as

243
00:13:00,529 --> 00:13:03,980
this, then there would be a solution, where you could just

244
00:13:03,980 --> 00:13:04,789
scale those.

245
00:13:04,789 --> 00:13:08,099
Actually, there would be an infinite number of solutions

246
00:13:08,100 --> 00:13:09,500
in terms of x and y.

247
00:13:09,500 --> 00:13:13,580
But if the vector is slightly different, in terms of its

248
00:13:13,580 --> 00:13:15,320
direction, then there is no solution.

249
00:13:15,320 --> 00:13:18,310
There is no combination of this vector and this vector

250
00:13:18,309 --> 00:13:19,979
that can add you up to this one.

251
00:13:19,980 --> 00:13:22,384
And it's something for you think about a little bit.

252
00:13:22,384 --> 00:13:23,360
It might be obvious to you.

253
00:13:23,360 --> 00:13:24,730
But another way to think about it is, when you're trying to

254
00:13:24,730 --> 00:13:29,000
take sums of vectors, any other vector, in order to move

255
00:13:29,000 --> 00:13:31,149
it in that direction, you have to have a little bit of one

256
00:13:31,149 --> 00:13:33,470
direction and a little bit of another direction, to get to

257
00:13:33,470 --> 00:13:34,230
any other vector.

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And if both of your ingredient vectors are the same

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direction, there's no way to get to a different one.

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Anyway, I'm probably just being circular in what I'm

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explaining.

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But that hopefully gives you a little bit of an intuition of

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well, one, you now know what a singular matrix is.

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You know when you can not find its inverse.

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You know that when the determinant is 0, you won't

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find an inverse.

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And hopefully-- and this was the whole point of this

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video-- you have an intuition of why that is.

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Because if you're looking at the vector problem, there's no

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way that you can find-- that either there's no solution to

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finding the combination of the vectors that get you to that

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vector, or there are an infinite number.

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And the same thing is true of finding the

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intersection of two lines.

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They're either parallel, or they're the same line, if the

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determinant is 0.

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Anyway, I will see you in the next video.

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