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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
Inverting Matrices (part 3) Using Gauss-Jordan elimination to invert a 3x3 matrix.
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- I will now show you my preferred way of finding an
- inverse of a 3 by 3 matrix.
- And I actually think it's a lot more fun.
- And you're less likely to make careless mistakes.
- But if I remember correctly from Algebra 2, they didn't
- teach it this way in Algebra 2.
- And that's why I taught the other way initially.
- But let's go through this.
- And in a future video, I will teach you why it works.
- Because that's always important.
- But in linear algebra, this is one of the few subjects where
- I think it's very important learn how to do the operations
- first. And then later, we'll learn the why.
- Because the how is very mechanical.
- And it really just involves some basic arithmetic
- for the most part.
- But the why tends to be quite deep.
- So I'll leave that to later videos.
- And you can often think about the depth of things when you
- have confidence that you at least understand the hows.
- So anyway, let's go back to our original matrix.
- And what was that original matrix that I
- did in the last video?
- It was 1, 0, 1, 0, 2, 1, 1, 1, 1.
- And we wanted to find the inverse of this matrix.
- So this is what we're going to do.
- It's called Gauss-Jordan elimination, to find the
- inverse of the matrix.
- And the way you do it-- and it might seem a little bit like
- magic, it might seem a little bit like voodoo, but I think
- you'll see in future videos that it makes a lot of sense.
- What we do is we augment this matrix.
- What does augment mean?
- It means we just add something to it.
- So I draw a dividing line.
- Some people don't.
- So if I put a dividing line here.
- And what do I put on the other side of the dividing line?
- I put the identity matrix of the same size.
- This is 3 by 3, so I put a 3 by 3 identity matrix.
- So that's 1, 0, 0, 0, 1, 0, 0, 0, 1.
- All right, so what are we going to do?
- What I'm going to do is perform a series of elementary
- row operations.
- And I'm about to tell you what are valid elementary row
- operations on this matrix.
- But whatever I do to any of these rows here, I have to do
- to the corresponding rows here.
- And my goal is essentially to perform a bunch of operations
- on the left hand side.
- And of course, the same operations will be applied to
- the right hand side, so that I eventually end up with the
- identity matrix on the left hand side.
- And then when I have the identity matrix on the left
- hand side, what I have left on the right hand side will be
- the inverse of this original matrix.
- And when this becomes an identity matrix, that's
- actually called reduced row echelon form.
- And I'll talk more about that.
- There's a lot of names and labels in linear algebra.
- But they're really just fairly simple concepts.
- But anyway, let's get started and this should become a
- little clear.
- At least the process will become clear.
- Maybe not why it works.
- So first of all, I said I'm going to perform a bunch of
- operations here.
- What are legitimate operations?
- They're called elementary row operations.
- So there's a couple things I can do.
- I can replace any row with that row
- multiplied by some number.
- So I could do that.
- I can swap any two rows.
- And of course if I swap say the first and second row, I'd
- have to do it here as well.
- And I can add or subtract one row from another row.
- So when I do that-- so for example, I could take this row
- and replace it with this row added to this row.
- And you'll see what I mean in the second.
- And you know, if you combine it, you could you could say,
- well I'm going to multiple this row times negative 1, and
- add it to this row, and replace this row with that.
- So if you start to feel like this is something like what
- you learned when you learned solving systems of linear
- equations, that's no coincidence.
- Because matrices are actually a very good way to represent
- that, and I will show you that soon.
- But anyway, let's do some elementary row operations to
- get this left hand side into reduced row echelon form.
- Which is really just a fancy way of saying, let's turn it
- into the identity matrix.
- So let's see what we want to do.
- We want to have 1's all across here.
- We want these to be 0's.
- Let's see how we can do this efficiently.
- Let me draw the matrix again.
- So let's get a 0 here.
- That would be convenient.
- So I'm going to keep the top two rows the same.
- 1, 0, 1.
- I have my dividing line.
- 1, 0, 0.
- I didn't do anything there.
- I'm not doing anything to the second row.
- 0, 2, 1.
- 0, 1, 0.
- And what I'm going to do, I'm going to replace this row--
- And just so you know my motivation, my goal
- is to get a 0 here.
- So I'm a little bit closer to having the
- identity matrix here.
- So how do I get a 0 here?
- What I could do is I can replace this row with this row
- minus this row.
- So I can replace the third row with the third row
- minus the first row.
- So what's the third row minus the first row?
- 1 minus 1 is 0.
- 1 minus 0 is 1.
- 1 minus 1 is 0.
- Well I did it on the left hand side, so I have to do it on
- the right hand side.
- I have to replace this with this minus this.
- So 0 minus 1 is minus 1.
- 0 minus 0 is 0.
- And 1 minus 0 is 1.
- Fair enough.
- Now what can I do?
- Well this row right here, this third row, it has 0 and 0-- it
- looks a lot like what I want for my second row in the
- identity matrix.
- So why don't I just swap these two rows?
- Why don't I just swap the first and second rows?
- So let's do that.
- I'm going to swap the first and second rows.
- So the first row stays the same.
- 1, 0, 1.
- And then the other side stays the same as well.
- And I'm swapping the second and third rows.
- So now my second row is now 0, 1, 0.
- And I have to swap it on the right hand side.
- So it's minus 1, 0, 1.
- I'm just swapping these two.
- So then my third row now becomes what the
- second row was here.
- 0, 2, 1.
- And 0, 1, 0.
- Fair enough.
- Now what do I want to do?
- Well it would be nice if I had a 0 right here.
- That would get me that much closer to the identity matrix.
- So how could I get as 0 here?
- Well what if I subtracted 2 times row two from row one?
- Because this would be, 1 times 2 is 2.
- And if I subtracted that from this, I'll get a 0 here.
- So let's do that.
- So the first row has been very lucky.
- It hasn't had to do anything.
- It's just sitting there.
- 1, 0, 1, 1, 0, 0.
- And the second row's not changing for now.
- Minus 1, 0, 1.
- Now what did I say I was going to do?
- I'm going to subtract 2 times row two from row three.
- So this is 0 minus 2 times 0 is 0.
- 2 minus 2 times 1, well that's 0.
- 1 minus 2 times 0 is 1.
- 0 minus 2 times negative 1 is-- so let's remember 0 minus
- 2 times negative 1.
- So that's 0 minus negative 2, so that's positive 2.
- 1 minus 2 times 0.
- Well that's just still 1.
- 0 minus 2 times 1.
- So that's minus 2.
- Have I done that right?
- I just want to make sure.
- 0 minus 2 times-- right, 2 times minus 1 is minus 2.
- And I'm subtracting it, so it's plus.
- OK, so I'm close.
- This almost looks like the identity matrix or reduced row
- echelon form.
- Except for this 1 right here.
- So I'm finally going to have to touch the top row.
- And what can I do?
- well how about I replace the top row with the top row minus
- the bottom row?
- Because if I subtract this from that,
- this'll get a 0 there.
- So let's do that.
- So I'm replacing the top row with the top row
- minus the third row.
- So 1 minus 0 is 1.
- 0 minus 0 is 0.
- 1 minus 1 is 0.
- That was our whole goal.
- And then 1 minus 2 is negative 1.
- 0 minus 1 is negative 1.
- 0 minus negative 2., well that's positive 2.
- And then the other rows stay the same.
- 0, 1, 0, minus 1, 0, 1.
- And then 0, 0, 1, 2, 1, negative 2.
- And there you have it.
- We have performed a series of operations on
- the left hand side.
- And we've performed the same operations on
- the right hand side.
- This became the identity matrix, or
- reduced row echelon form.
- And we did this using Gauss-Jordan elimination.
- And what is this?
- Well this is the inverse of this original matrix.
- This times this will equal the identity matrix.
- So if this is a, than this is a inverse.
- And that's all you have to do.
- And as you could see, this took me half the amount of
- time, and required a lot less hairy mathematics than when I
- did it using the adjoint and the cofactors and the
- determinant.
- And if you think about it, I'll give you a little hint of
- why this worked.
- Every one of these operations I did on the left hand side,
- you could kind of view them as multiplying-- you know, to get
- from here to here, I multiplied.
- You can kind of say that there's a matrix.
- That if I multiplied by that matrix, it would have
- performed this operation.
- And then I would have had to multiply by another matrix to
- do this operation.
- So essentially what we did is we multiplied by a series of
- matrices to get here.
- And if you multiplied all of those, what we call
- elimination matrices, together, you essentially
- multiply this times the inverse.
- So what am I saying?
- So if we have a, to go from here to here, we have to
- multiply a times the elimination matrix.
- And this might be completely confusing for you, so ignore
- it if it is, but it might be insightful.
- So what did we eliminate in this?
- We eliminated 3, 1.
- We multiplied by the elimination matrix
- 3, 1, to get here.
- And then, to go from here to here, we've
- multiplied by some matrix.
- And I'll tell you more.
- I'll show you how we can construct
- these elimination matrices.
- We multiply by an elimination matrix.
- Well actually, we had a row swap here.
- I don't know what you want to call that.
- You could call that the swap matrix.
- We swapped row two for three.
- And then here, we multiplied by elimination
- matrix-- what did we do?
- We eliminated this, so this was row three,
- column two, 3, 2.
- And then finally, to get here, we had to multiply by
- elimination matrix.
- We had to eliminate this right here.
- So we eliminated row one, column three.
- And I want you to know right now that it's not important
- what these matrices are.
- I'll show you how we can construct these matrices.
- But I just want you to have kind of a leap of faith that
- each of these operations could have been done by multiplying
- by some matrix.
- But what we do know is by multiplying by all of these
- matrices, we essentially got the identity matrix.
- Back here.
- So the combination of all of these matrices, when you
- multiply them by each other, this must
- be the inverse matrix.
- If I were to multiply each of these elimination and row swap
- matrices, this must be the inverse matrix of a.
- Because if you multiply all them times
- a, you get the inverse.
- Well what happened?
- If these matrices are collectively the inverse
- matrix, if I do them, if I multiply the identity matrix
- times them-- the elimination matrix, this one times that
- equals that.
- This one times that equals that.
- This one times that equals that.
- And so forth.
- I'm essentially multiplying-- when you combine all of
- these-- a inverse times the identity matrix.
- So if you think about it just very big picture-- and I don't
- want to confuse you.
- It's good enough at this point if you just
- understood what I did.
- But what I'm doing from all of these steps, I'm essentially
- multiplying both sides of this augmented matrix, you could
- call it, by a inverse.
- So I multiplied this by a inverse, to get to the
- identity matrix.
- But of course, if I multiplied the inverse matrix times the
- identity matrix, I'll get the inverse matrix.
- But anyway, I don't want to confuse you.
- Hopefully that'll give you a little intuition.
- I'll do this later with some more concrete examples.
- But hopefully you see that this is a lot less hairy than
- the way we did it with the adjoint and the cofactors and
- the minor matrices and the determinants, et cetera.
- Anyway, I'll 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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