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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 2) Inverting a 3x3 matrix
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- We will now embark on what is probably my least favorite
- exercise or computation in mathematics-- and I think
- you'll see why-- where we will invert a 3 by 3 matrix.
- And in my mind, the only thing less pleasant than inverting a
- 3 by 3 matrix is inverting a 4 by 4 matrix.
- It very quickly becomes obvious to you that it's
- probably better for a computer to do this.
- But you need to learn how to do it.
- And it's a good exercise for me to do.
- And if I keep doing it my whole life, it'll prevent my
- brain from degrading.
- But as you'll see, this is almost an exercise in not
- making careless mistakes.
- So let's start with a 3 by 3 matrix and
- try to take the inverse.
- So let's say I have matrix A.
- I think I'm going to need a lot of space here, so I'll try
- to do this small, without being confusing.
- Matrix A.
- Let's say it's 1, 0-- and I'm specifically choosing this
- matrix because the numbers are reasonably non-hairy-- 0, 2,
- 1, 1, 1, 1.
- So the first thing that I do when I take an inverse of a 3
- by 3 matrix, I create what I call-- or not what I call,
- what everyone calls-- a matrix of minors.
- So let me write that down.
- Matrix of minors.
- Let me draw that out.
- So it's going to be another 3 by 3 matrix.
- And what it is, so this element, this top left
- element, is essentially going to be the determinant.
- If I were to take my original matrix, and I were to cross
- out this position's row and column.
- So for example, this 1, 1 position, row 1, column 1.
- I cross out row 1 and column 1.
- What numbers do I have left?
- I have this 2, 1, 1, 1.
- I have this right here.
- So it's the determinant of 2, 1, 1, 1.
- And actually maybe I'll write that down.
- So it's the determinant of 2, 1, 1, 1.
- So I'm gonna run out of space I'm sure.
- It's going to be 2, 1, 1, 1.
- The determinant.
- The absolute value sign says it's the determinant.
- Remember, all I did is, I said, OK, in position 1,1, let
- me cross out the column and the row 1,1, and take the
- determinant of what's left.
- Or the minor of this matrix.
- And then I will take the determinant-- so when I go to
- this position, I'm in row 1, column 2.
- I'm essentially going to take the determinant.
- If I were to cross out row 1 and column 2, what
- do I have left over?
- I have 0, 1, 1, 1.
- It might be confusing, but just remember-- I wish I had
- something I could cover this with.
- Unfortunately my fingers can't show up on this video.
- But if you cross this row and this column out, you're just
- left with this 0, this 1, this 1, and this 1.
- And you take the determinant of that minor.
- And we'll keep going.
- I'm probably going to run out of space here, but I will try
- my best. And so when you go to this position-- row 1, column
- 3-- what do you do?
- Well you cross out row 1, column 3.
- And then the determinant, or the minor that you have to do,
- is 0, 2, 1, 1.
- So the determinant of that two by two matrix.
- And then you keep doing that, so forth and so on.
- And I'm going to run out of space.
- But what I'm going to do is I'm just going
- to calculate it.
- I think you understand how to do it.
- Well you don't understand how to do it, but I think when we
- calculate it it'll make a little more sense.
- Let me actually just calculate it out.
- Because if I were to write these 2 by 2 matrices, I would
- run out of space.
- But anyway, let's go back to this position 1, 1.
- Cross out the first row, first column, I want the determinant
- of this thing right here.
- So what's the determinant of this 2 by 2?
- That's not too hard.
- It's 2 times 1, minus 1 times 1.
- So what's 2 times 1, minus 1 times 1?
- Well it's just 1.
- Then when we go to row 1, column 2, I want the
- determinant of 0, 1, 1, 1.
- So it's 0 times 1, minus 1 times 1.
- So 0 times 1 is 0, minus 1 times 1 is minus 1.
- And that's just this determinant right here.
- I'm just kind of reshowing you how I visualize when I cross
- out the rows and columns.
- So it's 0 times 1, minus 1 times 1.
- And in this position of course, you cross out this
- row, this column, and 0 times 1 minus 1 times 2.
- So that's minus 2.
- Let's keep going.
- All right.
- So now, when we're in row 2, column 1, we cross out row 2,
- cross out column 1.
- And we're left with this 0, this 1, this 1 and this 1.
- So it's 0 times 1, which is 0.
- Minus 1 times 1.
- So we're at minus 1.
- Then when we get row 2, column 2, we cross those two out, and
- we take the matrix of the minor that's left.
- So that's 1 times 1, minus 1 times 1.
- So that's 0.
- Almost. We're halfway done.
- OK, so then we're in row 2, column 3.
- So we cross out row 2, column 3.
- And what we have left is 1 times 1, minus 1 time 0.
- So that is just 1.
- Last row.
- OK, so we're in row 3, column 1.
- So we cross out row 3, column 1.
- You're left with 0 times 1 is 0.
- Minus 2 times 1.
- So that's minus 2.
- Then we're in row 3, column 2.
- So we cross out row 3, column 2.
- And you have 1 times 1, minus 0 times 1.
- So that's just 1.
- Last one.
- Row 3, column 3.
- So we cross out row 3, we cross out column 3.
- And you're just left with 1 times 2, minus 0 times 0.
- So that is 2.
- And if I haven't made any careless mistakes, that is our
- matrix of minors.
- Now we now have to convert this to what we call the
- matrix of cofactors.
- And actually this step is fairly straightforward.
- So to convert from a matrix of minors to a matrix of
- cofactors, you just have to remember this pattern.
- This pattern applies to any 3 by 3 matrix.
- Plus, minus, plus, minus, plus, minus,
- plus, minus, plus.
- And so you can kind of just imagine this as kind of a
- checkerboard of pluses and minuses.
- And you apply that to this.
- So what do I mean by that?
- Well that means, when you start, it's a checkerboard,
- and you start with a plus at the top left.
- And then you just keep alternating plus, minus.
- So if you applied this to this, you get
- the matrix of cofactors.
- Let me write that down.
- You can imagine this is a bit of a marathon of computation.
- OK, so the matrix of cofactors is essentially applying this
- pattern to the matrix of minors.
- So what do you do?
- You say this plus 1 times 1 is 1.
- But now we have a minus.
- So that's minus times minus 1 is positive 1.
- Then you have plus times minus 2 is minus 2.
- Then you have a minus here.
- Minus times minus 1 is positive 1.
- Plus times 0 is still 0.
- Minus times 1 is minus 1.
- Plus times minus 2 is minus 2.
- Minus applied to 1 is minus 1.
- And then plus applied to 2 is just 2.
- And we have our matrix of cofactors.
- And we are more than halfway done with
- inverting this matrix.
- And I just want to take a note here.
- What we're doing is kind of just a magic formula.
- It might seem a little bit like voodoo for you.
- But I just want you to keep in mind that in future videos, I
- will show you where this comes from.
- Although it will be quite hairy to prove
- it for a 3 by 3.
- But I'll definitely show it to you for a 2 by 2.
- And actually, I'll show you other algorithms that might
- make a little bit more intuitive sense for doing it
- for a 3 by 3.
- But I just wanted to show you how to do it this way, so that
- at least when you see it on your Algebra 2 exam-- because
- I think they actually teach this in Algebra 2-- you could
- at least, if the teacher asks you, solve for the matrix of
- minors or the cofactors or solve for the determinant of
- the inverse, you can do it.
- And then we'll worry about getting the intuition, which
- is not how I normally like to teach things.
- But this is an exception.
- But anyway, back to the problem.
- This is the matrix of cofactors.
- Now from this, we take the adjoint of matrix a-- or I
- learned from Wikipedia, the correct term is the adjugate
- of matrix a.
- But this is determined the notation is the adjugate of a.
- And all this is is the transpose of
- the matrix of cofactors.
- And I know I'm throwing out a lot of weird terminology here.
- But the transpose, all that means is that you switch the
- rows and the columns.
- So this one right here is in row 1, column 1.
- But you know, so the rows and columns are the same, so that
- just stays the same.
- So actually anything on the diagonal stays the same.
- Because this is row 2, column 2.
- This is row 3, column 3.
- So the diagonals stay the same.
- And then you switch places.
- You kind of flip across the diagonal.
- And what do I mean by that?
- Well this 1 was in row 1, column 2.
- So it'll then be moved to row 2, column 1.
- So this 1 right here will go here.
- So you can kind of say that it flipped over the diagonal.
- And similarly, this right here is in row 1, column 3.
- It's going to be switched to row 3, column 1.
- So it's going to go here.
- And you can kind of see that it just flipped over that end.
- So this minus 2 isn't this one.
- It's this one over here.
- And actually, we see that this matrix is symmetric.
- When you flip it, you actually get the same thing.
- So maybe it was a bad example.
- But I want you to understand that the transpose is where--
- if something like this number, if it's in a row 1, column 2,
- then it moves to row 2, column 1.
- So you're switching the rows and columns.
- But anyway, we could keep doing that.
- But essentially you're just flipping over the diagonal.
- So let's see.
- So then this number will be flipped to this position, so
- it goes there.
- This is in row 2, column 1.
- So it will go to column 2, row 1, which is that.
- And then if we go here, that's going to be flipped down here,
- flipped across the diagonal.
- So that's minus 1.
- This is going to be flipped all the way up there.
- So that's minus 2.
- And then this will be flipped there.
- This is minus 1.
- We are almost done.
- So this is the adjoint of matrix a.
- So to get the inverse of a-- and let me actually erase some
- of this, because we're going to run out of space otherwise.
- And as you can see, I'll be very impressed if I have not
- introduced a careless mistake yet.
- So let me erase all of this.
- I'm building an appetite just doing this problem.
- It's so taxing on me.
- So the inverse of matrix a is equal to 1 over the
- determinant of a times the adjugate, or
- adjoint, of matrix a.
- So we solved for this part.
- So now let's solve for the determinant.
- So the determinant of a-- and I kept the matrix of cofactors
- here for a reason-- the determinant of a is-- if you
- go across-- you can actually go across any row-- but just
- for simplicity, just remember it this way.
- You go across the top row, and you multiply each term times
- its corresponding cofactor, and you add them.
- So in this case, it'll be 1 times its corresponding
- cofactor, which is 1.
- Plus 0 times its corresponding cofactor, which is 1.
- Plus 1 times its corresponding cofactor plus minus 2.
- So this is 1 plus 0 minus 2.
- It equals minus 1.
- And thank God it was a relatively straightforward
- determinant.
- And if you didn't have this matrix of cofactors, the other
- way you could think about it-- and this is good because it
- gives you an intuition of how we even got to the matrix of
- cofactors-- you could view this as the same thing as 1
- times the determinant of its minor.
- So if you cross out the row and the column, it's this
- determinant.
- So it's 2, 1, 1, 1.
- And remember there was that pattern.
- You have plus, and then you go minus.
- So minus 0 times the determinant of its minor.
- So you cross out that row, that column.
- So 0, 1, 1, 1.
- And then we switch again.
- We go back to plus.
- Plus 1 times the determinant of its minor.
- So you cross out that row, that column.
- You get 0, 2, 1, 1.
- And you could compute this out.
- And this is this cofactor.
- This, with a minor sign, this is just a minor.
- And then when you apply the minus sign, it
- becomes this cofactor.
- And then this is that minor.
- And since it's a plus sign there, that's that cofactor.
- But anyway, I just wanted to explain that, and hopefully it
- hasn't confused you.
- But we're ready now to solve the inverse of a.
- We know that the determinant of a is equal to minus 1.
- We know that the adjugate of a is this number here.
- So we now can solve for the inverse.
- And let's do that.
- Let me erase all of this stuff.
- Cause actually, after I solve for the inverse, I want to
- prove to you that it is the inverse-- maybe.
- If I have enough time.
- Because I just realized I'm running pretty long.
- That might be a good exercise for you.
- OK.
- So the inverse of a is equal to 1 over the determinant.
- We figured out the determinant is negative 1 times the
- adjugate of a.
- 1, 1, minus 2.
- 1, 0, minus 1.
- Minus 2, minus 1, 2.
- So this is just minus 1, right?
- So we just apply minus 1 times everything.
- So we get-- if I haven't made any careless mistakes-- minus
- 1, minus 1, plus 2, minus 1, 0, 1, 2, 1, minus 2.
- I think that I have-- let's see, I just did a minus times
- everything.
- That looks right.
- And so that is a inverse.
- And it only took us 17 minutes.
- And I will leave you there, because it will probably take
- me another 5 or 10 minutes to even prove,
- but that might be a good exercise for you --
- to multiply this matrix times this matrix, and make sure
- that you get the identity matrix.
- I will see you in the next video.
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