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- In the last video we defined the notion of a determinant of
- a 2 by 2 matrix.
- So if I have some matrix-- let's just call it B-- if my
- matrix B looks like this, if its entries are a, b, c, d,
- we've defined to determinant of B.
- Which could also be written as B with these lines around it,
- which could also be written as the entries of the matrix with
- those lines around it, a, b, c, d.
- And I don't want you to get these confused.
- This is the matrix when you have the brackets.
- This is the determinant of the matrix, when you just have
- these straight lines.
- And this, by definition, was equal to ad minus bc.
- And you saw in the last video, or maybe you saw in the last
- video, what the motivation for this came from.
- When we figured out the inverse of B, we determined
- that it was equal to 1 over ad minus bc times another matrix,
- which was essentially these two entry swaps, you
- got a d and an a.
- And then these two entries made negative, so
- minus c and minus b.
- This was the inverse of b.
- And we said, well, when is this defined?
- This is defined as long as this character right here does
- not equal 0.
- So you said hey, this looks pretty important.
- Let's call this thing right there the determinant.
- And then we could say that B is invertible, if and only if,
- the determinant of B does not equal 0.
- Because if it equals 0, then this formula for your inverse
- won't be well defined.
- And we just got this from our technique of creating an
- augmented matrix whatnot.
- But the big take away is we defined this notion of a
- determinant it for a 2 by 2 matrix.
- Now the next question is, well that's just a 2 by 2,
- everything we do in linear algebra, we like to generalize
- it to higher numbers of rows and columns.
- So the next step, at least-- let's just do baby steps--
- let's start with a 3 by 3.
- Let's define what its determinant is.
- So let me construct a 3 by 3 matrix here.
- Let's say my matrix A is equal to-- let me just write its
- entries-- first row, first column, first row, second
- column, first row, third column.
- Then you have a2 1, a2 2, a2 3.
- Then you have a3 1, third row first column, a3
- 2, and then a3 3.
- That is a 3 by 3 matrix.
- Three rows and three columns.
- This is 3 by 3.
- I am going to define the determinant of A.
- So this is a definition.
- I'm going to define the determinant of this 3 by 3
- matrix A as being equal to-- and this is a little bit
- convoluted, but you'll get the hang of it eventually.
- In the next several videos we're just going to do a ton
- of determinants.
- So it just becomes a bit of second nature to you.
- It's a little computationally intensive sometimes.
- But it equals this first row.
- It equals a1 1 times the determinant of the matrix you
- get, if you get rid of this guy's column and row.
- So if you get rid of this guy's column and row, you're
- left with this matrix here.
- So times the determinant of the matrix a2 2, a2 3, a3 2,
- and then a3 3.
- Just like that.
- So that's our first entry and that's a plus this.
- And then I said it's a plus this, because the next entry's
- going to be a minus.
- You have a minus this guy right here.
- So then you're going to have minus a1 2 times the matrix
- you get if you eliminate his column and his row.
- So times, you're going to get these entries right there.
- So a2 1, a2 3, a3 1, and then you have a3 3.
- We're not quite done.
- You could probably guess with the next one's going to be.
- Then you're going to have a plus-- let me switch to a
- better color-- plus this guy.
- Plus a1 3 times the determinant of its-- I guess
- you could call it-- its sub-matrix.
- We'll call it that for now.
- So this matrix right here.
- So a2 1, a2 2, a3 1, a3 2.
- This is our definition of the
- determinant of a 3 by 3 matrix.
- And the motivation is, because when you take the determinant
- of a 3 by 3 it turns out-- I haven't shown it to you yet--
- that the property is the same.
- That if the determinant of this is 0, you will not be
- able to find an inverse.
- And when I defined determinant in this way.
- If the determinant does not equal 0, you will be able to
- find an inverse.
- So that's where this came from.
- And I haven't shown you that yet.
- And I might not show you because it's super
- computational.
- It'll take a long time.
- It'll be very hairy and I'll make careless mistakes.
- But the motivation comes from the exact same place as the 2
- by 2 version.
- But I think what you probably want to see right now is at
- least just this thing applied to an actual matrix, because
- this looks all abstract right now.
- But if we do it with an actual matrix, you'll actually see
- it's not too bad.
- So let's leave the definition up there, and let's say that I
- have the matrix 1, 2, 4, 2, 2, minus 1, 3, and 4, 0, 1.
- So by our definition of a determinant, the determinant
- of this guy right here-- so let's say I call that matrix
- C-- C is equal to that.
- So if we want to figure out the determinant of C, the
- determinant of C is equal to-- I take this guy right here,
- let me take that 1-- times the determinant of-- let's just
- call it the submatrix, right here.
- So we have a minus 1, we have a 3, we have a 0,
- and we have a 1.
- Just like that.
- Notice, I got rid of this guy's column
- and this guy's row.
- And I was just left with minus 1, 3, 0, 1.
- Next, I take this guy.
- And this is the trick.
- You have to alternate signs.
- If you start with a positive here, this next one's going to
- be a minus.
- So you're going to have minus 2 times the submatrix-- we can
- call it-- if we get rid of this guy's column
- and this guy's row.
- So 2, 3, 4, 1.
- I just blanked this out.
- If I could videotape my finger, I would cover my
- finger over this column right here and over that row, and
- you'd just see a 2, a 3, a 4, and a 1.
- And that's what I put right there.
- And then finally, we went plus, minus, plus.
- So finally, we'll have plus 4 times the determinant of the
- submatrix, if you get rid of that row in that column.
- So 2, minus 1, 4, 0.
- Now, these are pretty straightforward.
- These are not too bad to compute.
- Let's actually do it.
- So this is going to be equal to 1 times what?
- Minus 1 times 1.
- Let me just write it out.
- Minus 1 times 1, minus 0 times 3.
- This just comes from the definition of a 2 by 2
- determinant.
- We've already defined that.
- And then we're going to have a minus 2 times 2 times 1,
- minus 4 times 3.
- And then finally, we're going to have a plus 4 times 2 times
- 0 minus minus 1 times 4.
- I wrote it all out so you can see.
- This thing right here is just this thing right here.
- And then you have the 4 out front.
- This thing right here was just this thing right here.
- So it's the determinant of the 2 by 2 submatrix for each of
- these guys.
- And if we compute this, this is equal to-- minus 1 times 1
- is minus 1.
- Minus 0, that's 0.
- So this is a minus 1 times 1, so that's a minus 1.
- And then we get-- what is this equal to?
- This right here is 12.
- So you get 2 minus 12.
- Right?
- You get 2 times 1 minus 4 times 3.
- So it's minus 10.
- So that is equal to minus 10.
- And then you have a minus 10 times a minus 2.
- So that becomes a plus 20, right?
- Minus 2 times minus 10.
- And then finally, in the green, we have 2 times 0,
- that's just a 0.
- And then you have minus 1 times 4, which is minus 4.
- Then you have a minus sign here, so it's plus 4.
- So this all becomes a plus 4.
- Plus 4 times 4 is 16, so plus 16.
- And what do we get when we add this up?
- We get 20 plus 16 minus 1.
- It is equal to 35.
- We're done.
- We found the determinant of our 3 by 3 matrix.
- Not too bad.
- Right there, so that is equal to the determinant of C.
- So the fact that this isn't 0 tells you that C is
- invertible.
- In the next video, we'll try to extend this to n by n
- square matrices.

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