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Idea Behind Inverting a 2x2 Matrix
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Finding the Determinant of a 2x2 matrix
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Determinant of a 2x2 matrix
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Inverse of a 2x2 matrix
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Inverse of a 2x2 matrix
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Matrices to solve a system of equations
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Matrices to solve a vector combination problem
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Finding the determinant of a 3x3 matrix method 1
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Finding the determinant of a 3x3 matrix method 2
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Determinant of a 3x3 matrix
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Inverting 3x3 part 1: Calculating Matrix of Minors and Cofactor Matrix
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Inverting 3x3 part 2: Determinant and Adjugate of a Matrix
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Inverse of a 3x3 matrix
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Inverting matrices (part 2)
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Inverting Matrices (part 3)
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Singular Matrices
Singular Matrices When and why you can't invert a matrix.
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- Perhaps even more interesting than finding the inverse of a
- matrix is trying to determine when an inverse of a matrix
- doesn't exist. Or when it's undefined.
- And a square matrix for which there is no inverse, of which
- an inverse is undefined is called a singular matrix.
- So let's think about what a singular matrix will look
- like, and how that applies to the different problems that
- we've address using matrices.
- So if I had the other 2 by 2, because that's
- just a simpler example.
- But it carries over into really any size square matrix.
- So let's take our 2 by 2 matrix.
- And the elements are a, b, c and d.
- What's the inverse of that matrix?
- This hopefully is a bit of second nature to you now.
- It's 1 over the determinant of a, times the adjoint of a.
- And in this case, you just switch these two terms. So you
- have a d and an a.
- And you make these two terms negative.
- So you have minus c and minus b.
- So my question to you is, what will make this entire
- expression undefined?
- Well it doesn't matter what numbers I have. If I have
- numbers here that make a defined, then I can obviously
- swap them or make them negative, and it won't change
- this part of the expression.
- But what would create a problem is if we attempted to
- divide by 0 here.
- If the determinant of the matrix A were undefined.
- So A inverse is undefined, if and only if-- and in math they
- sometimes write it if with two f's-- if and only if the
- determinant of A is equal to 0.
- So the other way to view that is, if a determinant of any
- matrix is equal to 0, then that matrix is a singular
- matrix, and it has no inverse, or the inverse is undefined.
- So let's think about in conceptual terms, at least the
- two problems that we've looked at, what a 0 determinant
- means, and see if we can get a little bit of intuition for
- why there is no inverse.
- So what is a 0 determinant?
- In this case, what's a determinant of this 2 by 2?
- Well the determinant of A is equal to what?
- It's equal to ad minus bc.
- So this matrix is singular, or it has no inverse, if this
- expression is equal to 0.
- So let me write that over here.
- So if ad is equal to bc-- or we can just manipulate things,
- and we could say if a/b is equal to c/d-- I just divided
- both sides by b, and divided both sides by d-- so if the
- ratio of a:b is the same as the ratio of c:d, then this
- will have no inverse.
- Or another way we could write this expression, if a/c-- if I
- divide both sides by c, and divide both sides by d-- is
- equal to b/d.
- So another way that this would be singular is if-- and it's
- actually the same way.
- If this is true, then this is true.
- These are the same.
- Just a little bit of algebraic manipulation.
- But if the ratio of a:c is equal to the ratio of b:d, and
- you can think about why that's the same thing.
- The ratio of a:b being the same thing as
- the ratio of c:d.
- But anyway, I don't want to confuse you.
- But let's think about how that translates into some of the
- problems that we looked at.
- So let's say that we wanted to look at the problem-- Let's
- say that we had this matrix representing the linear
- equation problem.
- Well, actually, this would be either one.
- So I have a, b, c, d times x, y Is equal to two other
- numbers that we haven't used yet, e and f.
- So if we have this matrix equation representing the
- linear equation problem, then the linear equation problem
- would be translated a times x plus b times y is equal to e.
- And c times x plus d times y is equal to f.
- And we would want to see where these two intersect.
- That would be the solution, the vector
- solution to this equation.
- And so, just to get a visual understanding of what these
- two lines look like, let's put it into the
- slope y-intercept form.
- So this would become what?
- In this case, y is equal to what?
- y is equal to minus a/b, x plus e/b.
- I'm just skipping some steps.
- But you subtract ax from both sides.
- And then divide both sides by b, and you get that.
- And then this equation, if you put it in the same form, just
- solve for y.
- You get y is equal to minus c/d x plus f/y.
- So let's think about this.
- I should probably change colors because it looks too--
- Let's think about what these two equations would look like
- if this holds.
- And we said if this holds, then we have no determinant,
- and this becomes a singular matrix, and it has no inverse.
- And since it has no inverse, you can't solve this equation
- by multiplying both sides by the inverse, because the
- inverse doesn't exist.
- So let's think about this.
- If this is true, we have no determinant, but what does
- that mean intuitively in terms of these equations?
- Well if a/b is equal to c/d, these two lines will have the
- same slope.
- They'll have the same slope.
- So if these two expressions are different, then what do we
- know about them?
- If two lines that have the same slope and different
- y-intercepts, they're parallel to each other, and they will
- never, ever intersect.
- So let me draw that, just so you get the-- this top line--
- They don't have to be positive numbers, but since this has a
- negative, I'll draw it as a negative slope.
- So that's the first line.
- And its y-intercept will be e/b.
- That's this line right here.
- And then the second line-- let me do it in another color-- I
- don't know if it's going to be above or below that line, but
- it's going to be parallel.
- It'll look something like this.
- And that line's y-intercept-- so that's this line-- that
- line's y intercept is going to be f/y.
- So if e/b and f/y are different terms, but both
- lines have the same equation, they're going to be parallel
- and they'll never intersect.
- So there actually would be no solution.
- If someone told you-- just the traditional way that you've
- done it, either through substitution, or through
- adding or subtracting the linear equations-- you
- wouldn't be able to find a solution where these two
- intersect, if a/b is equal to c/d.
- So one way to view the singular matrix is that you
- have parallel lines.
- Well then you might say, hey Sal, but these two lines would
- intersect if e/b equaled f/y.
- If this and this were the same, then these would
- actually be the identical lines.
- And not only would they intersect, they would
- intersect in an infinite number of places.
- But still you would have no unique solution.
- You'd have no one solution to this equation.
- It would be true at all values of x and y.
- So you can kind of view it when you apply the matrices to
- this problem.
- The matrix is singular, if the two lines that are being
- represented are either parallel, or they are the
- exact same line.
- They're parallel and not intersecting at all.
- Or they are the exact same line, and they intersect at an
- infinite number of points.
- And so it kind of makes sense that the A
- inverse wasn't defined.
- So let's think about this in the context of the linear
- combinations of vectors.
- That's not what I wanted to use to erase it.
- So when we think of this problem in terms of linear
- combination of factors, we can think of it like this.
- That this is the same thing as the vector ac times x plus the
- vector bd times y, is equal to the vector ef.
- So let's think about it a little bit.
- We're saying, is there some combination of the vector ac
- and the vector bd that equals the vector ef.
- But we just said that if we have no inverse here, we know
- that because the determinant is 0.
- And if the determinant is 0, then we know in this situation
- that a/c must equal b/d.
- So a/c is equal to b/d.
- So what does that tell us?
- Well let me draw it.
- And maybe numbers would be more helpful here.
- But I think you'll get the intuition.
- I'll just draw the first quadrant.
- I'll just assume both sectors are in the first quadrant.
- Let me draw.
- The vector ac.
- Let's say that this is a.
- Let me do it in a different color.
- So I'm gonna draw the vector ac.
- So if this is a, and this is c, then the vector
- ac looks like that.
- Let me draw it.
- I want to make this neat.
- The vector ac is like that.
- And then we have the arrow.
- And what would the vector bd look like?
- Well the vector bd-- And I could draw
- it arbitrarily someplace.
- But we're assuming that there's no derivative-- sorry,
- no determinant.
- Have I been saying derivative the whole time?
- I hope not.
- Well, we're assuming that there's no
- determinant to this matrix.
- So if there's no determinant, we know that
- a/c is equal to b/d.
- Or another way to view it is that c/d is equal to d/b.
- But what that tells you is that both of these vectors
- kind of have the same slope.
- So if they both start at point 0, they're going to go in the
- same direction.
- They might have a different magnitude, but they're going
- to go in the same direction.
- So if this is point b, and this is point d, vector bd is
- going to be here.
- And if that's not obvious to you, think a little bit about
- why these two vectors, if this is true, are going to point in
- the same direction.
- So that vector is going to essentially overlap.
- It's going to have the same direction as this vector, but
- it's just going to have a different magnitude.
- It might have the same magnitude.
- So my question to you is, vector ef, we don't know where
- vector ef is.
- Well let's pick some arbitrary point.
- Let's say that this is e, and this is f.
- So this is vector ef up there.
- Let me do it in a different color.
- Vector ef, let's say it's there.
- So my question to you is, if these two vectors are in the
- same direction.
- Maybe of different magnitude.
- Is there any way that you can add or subtract combinations
- of these two vectors to get to this vector?
- Well no, you can scale these vectors and add them.
- And all you're going to do is kind of move along this line.
- You can get to any other vector.
- There's a multiple of one of these vectors.
- But because these are the exact same direction, you
- can't get to any vector that's in a different direction.
- So if this vector is in a different direction, there's
- no solution here.
- If this vector just happened to be in the same direction as
- this, then there would be a solution, where you could just
- scale those.
- Actually, there would be an infinite number of solutions
- in terms of x and y.
- But if the vector is slightly different, in terms of its
- direction, then there is no solution.
- There is no combination of this vector and this vector
- that can add you up to this one.
- And it's something for you think about a little bit.
- It might be obvious to you.
- But another way to think about it is, when you're trying to
- take sums of vectors, any other vector, in order to move
- it in that direction, you have to have a little bit of one
- direction and a little bit of another direction, to get to
- any other vector.
- And if both of your ingredient vectors are the same
- direction, there's no way to get to a different one.
- Anyway, I'm probably just being circular in what I'm
- explaining.
- But that hopefully gives you a little bit of an intuition of
- well, one, you now know what a singular matrix is.
- You know when you can not find its inverse.
- You know that when the determinant is 0, you won't
- find an inverse.
- And hopefully-- and this was the whole point of this
- video-- you have an intuition of why that is.
- Because if you're looking at the vector problem, there's no
- way that you can find-- that either there's no solution to
- finding the combination of the vectors that get you to that
- vector, or there are an infinite number.
- And the same thing is true of finding the
- intersection of two lines.
- They're either parallel, or they're the same line, if the
- determinant is 0.
- Anyway, I will see you in the next video.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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