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Ok, this is Lecture 20
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and this is the final lecture
on determinant
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And it's about the application.
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So we work hard in the
last two lectures
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to get a formula for the determinant
and the properties of the determinants
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Now to use the determinant and...
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and always this determinant packs
all the information into a...
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into a single number
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And that number can give us
formulas for all sorts of things
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that we've been calculating
already without formulas
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Now what was A inverse?
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So I'm beginning
with a formula for A inverse.
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2*2 formula we know, right?
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The 2*2 formula for A inverse
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the inverse of a, b, c, d inverse is one
over the determinant times d, a, -b, -c
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Somehow, I want to see
what's going on for 3*3 and n*n.
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And actually, maybe you can see
what's going on from this 2*2 case.
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So there is a formula for the inverse
and what could I divide by?
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That't the determinant.
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So what I am looking for is a formula
where it has 1 over the determinant
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and you remember
why that makes good sense
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Because then that perfect as long as
the determinant isn't a zero
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and that's exactly
when there is an inverse.
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But now I have to ask can we
recognize any of this stuff?
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Do you recognize
what that number d is from the past?
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From the last, from the last lecture
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My hint is think cofactors.
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Because my formula is going to be
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my formula for the inverse is going
to be one over the determinant
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times a matrix of a cofactor.
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So remember that d,
what's that a cofactor of?
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Remember the cofactors,
if,that's the 1,1 cofactor.
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Because if I strike out row
and column one, I'm left with d.
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And what's -b?
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Ok, which cofactor is that one?
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Oh, -b is the cofactor of c, right?
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If I strike out the c
I'm left with a b there
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and why there's a minus sign?
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Because the c was in a 2,1 position
and 2 plus 1 is odd.
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So there was a minus in the cofactor
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And that's it.
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Ok, I will write down next
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What's my formula is
here's the big formula
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For the a, for A inverse.
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It's one over the determinant of a
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and then some matrix
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And that matrix is
the matrix of cofactors.
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C
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only one thing I, it turns,
you'll see, I have to, I transpose it
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So this is the matrix of cofactors
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But I'll just, but why don't we just
call that a cofactor matrix?
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So the 1,1 entry of c
is the 1,1 cofactor
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the thing we get by throwing away
the row and column one, it's the d.
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And, er, because of the transpose
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what I see up here is the cofactors
of this guy down here, right?
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That's where the transpose come in.
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What I see here,
this is the cofactor not of this one
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because I've transposed
this is the cofactor of the b
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When I throw away the b
the b row and the b column
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I'm left with this c
and I have that minus sign again
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And of course, the 2,2 entry is
the cofactor of d, and that is a
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Ok, so there is the formula
but we've got to think why?
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I mean, it works in the 2*2 case
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but a lot of other formulas
just work as well
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We have to see why that's true.
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In other words, why is the...
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so this is what I'm aimed to find
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and let's just, so look to see
what is that telling us
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That tells us the formula...
the expression for A inverse.
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Let's look at 3*3
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Now I just write down
a, b, c, d, e, f, g, h, i
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and I'm looking for its inverse.
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And what kind of formula
do I see there?
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The determinant is a bunch of
the products of three factors, right?
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The determinant of this
matrix involves a, e, i
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and bf times g, and c times d times h
and minus c*e*g, and so on
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So things with three factors
so in here
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things with...
how many fa...
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how many factors that the things
in the cofactor matrix had
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What's a typical cofactor?
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What's the cofactor of a?
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The cofactor of a, the 1,1 entry up here
in the inverse, is...
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I throw away the row
and column containing a
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and I take the determinant of
what's left, that's the cofactor.
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And that's e*i minus f*h.
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Products of the two things
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that I'm just making
the observation that
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the determinant of a involves
products of n entries
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and, then the cofactors matrix
involves products of n-1 entries
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and I, we never notice any
of this stuffs
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when they were computing the
inverse by the Gauss Jordan method
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whatever you remember how we did it.
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We took the matrix a
we tuck the identity next to it
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we did the elimination
so a became the identity
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and then the identity's
sum was the inverse
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Well that was great numerically.
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But, whenever, knew
what was going on basically
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and now we see
what the formula is in terms of letters
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What's the algebra
instead of the algorithm
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Ok, but I have to say
why this is right?
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Right? I see, this is...
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that's a pretty magic formula
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where does it come from?
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Well, I'll just check it.
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Having, having gotten up there
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Let me, I'll, I'll say,
how can we check?
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What do I want to check?
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I want to check that A times
this inverse gives the identity.
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So I want to check that A
times this thing, A times this
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Now I'm gonna write the inverse
gives the identity.
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So I'll check that A times C transpose
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Let me bring the determinant up here
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Determinant of A times the identity
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That's my job, that's it.
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That, if this is true, and it is
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then I correctly identify
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A inverse and C transpose
divided by the determinant.
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Ok, but why is this true?
Why is that true?
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Let me, let me put down
what I'm doing here.
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I have an A, here's a, here is a11
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I'm doing this multiplication
along to a1n.
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And then down this side asteroid
will be an1 along to ann.
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I'm multiplying by the
cofactor matrix transpose.
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So when I transpose
it will be c11, c12 down to c1n.
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Notice, usually that one coming first
would mean, I mean, row one
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but I've transposed
so those were the cofactors
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this first column are
the cofactors from row one.
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And then the left columns would
be the cofactors from row n.
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And why should that come out
to be anything good?
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In fact, why should it
come out to be as good as this.
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Well you can tell me what the 1,1 entry
in the product is.
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This is to see if you see the main point
if you just tell me one entry
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What do I get up there
in the 1,1 entry?
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When I do this row of, entry of...
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this row from A times
this column of cofactors
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What, what will I get there?
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Because we have seen this
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That we are building exactly on
what the last lecture reached.
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So this is a11 times c11,
a12 times c12, a1n times c1n.
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What dose that,
what does that sum up to?
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That's the co, that's the cofactor
formula for the determinant.
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That's this cofactor formula
which I wrote, which I,...
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We got last time that
the determinant of A is
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if I use row one,
let, let i equal one
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then I've a11 times this cofactor and
a12 times this cofactor and so on.
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And that gives me the determinant.
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And it works in this case.
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This row times this thing is,
sure enough, ad minus bc.
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but this one that it is
it always works.
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So up here, in this position
I'm getting the determinant of A.
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What about in the 2,2 position?
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Row two times column two there?
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What, what is that?
That's just the cofactors.
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That's just row two times
this cofactors
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So of course I get this
determinant again.
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And in the last here, this is the
last row, times this cofactors, exactly
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You see where we realize that the cofactor
formula is just the sum of product
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So of course we think, hey,
we've got a matrix multiplications there
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And we get the determinant of A
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Great.
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But there's one more
idea here, right?
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What else, what if I not...
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So I haven't got that formula
completely proved yet
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Because I still got to do all...
the all diagonal stuff
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which I want to be zero, right?
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I just want this to be the determinant
of A times the identity
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and then, then I'm a happy person
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So why should that be,
why should it be that
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one row times the cofactors from a
different row which becomes a column
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because I transpose, give zero.
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In other words, the cofactor formula
gives the determinant
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if the row and the cofactors...
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You know, if the entry is a and the
cofactors are from the same row
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But for some reason, if I take
the cofactors from the first...
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the entries from the first row, and
the cofactors from the second row
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for some reason
I will automatically get a zero
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And it's sort of like interesting
to say why does that happen?
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And can I just check, of course,
we know that it happens.
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And in this case, here're the
numbers from row one
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and here're the cofactors from
row two. Right?
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There was a number in row one
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and these were the cofactors
from row two.
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Because the cofactors of c is minus b
and the cofactors of d is a.
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and sure enough, that row times
this column gives...
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Please say it, zero, right.
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Ok, so, now how come?
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How come, have we been see
in the 2*2 case, why...
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Well I mean, I guess, we...
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00:13:55,570 --> 00:13:58,958
In one way, we certainly
do see it, because it's right here.
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00:13:58,993 --> 00:14:00,930
I mean we just do it,
now we get zero.
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00:14:00,965 --> 00:14:06,908
But we want to think of some reason
why the answer is zero.
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00:14:06,943 --> 00:14:09,855
Some reason that we
can use in the n*n case.
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00:14:10,208 --> 00:14:12,459
So, let's, here's, here's my thinking.
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We must be...
if we putting the answer of zero
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00:14:17,823 --> 00:14:21,993
we suspect that
what we're doing somehow
196
00:14:22,028 --> 00:14:27,126
we've taken some of the determinant
of some matrix that has two equal rows
197
00:14:27,969 --> 00:14:33,775
So I believe that if we multiply these
by the cofactors from some other row
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00:14:33,810 --> 00:14:36,710
We're taking the determinant
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00:14:37,326 --> 00:14:40,361
Yea, what matrix are we
taking this determinant of?
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00:14:40,396 --> 00:14:41,846
Here is, this is it.
201
00:14:41,881 --> 00:14:46,187
When we do that times this
we are really taking...
202
00:14:46,650 --> 00:14:49,473
Can I put this thing little
letters down here?
203
00:14:50,255 --> 00:14:57,229
I'm taking, let me look at
the matrix, a, b, a, b.
204
00:14:58,844 --> 00:15:05,100
Let me call that matrix As
A screwed up
205
00:15:05,135 --> 00:15:10,740
Ok, all right, so now that
matrix is certainly singular
206
00:15:10,775 --> 00:15:14,040
So if we sign this determinant
we're gonna get zero.
207
00:15:14,075 --> 00:15:19,056
But I claim that if we find this
determinant by the cofactor rule
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00:15:19,091 --> 00:15:21,279
go along the first row
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00:15:21,314 --> 00:15:24,656
we will take a times the
cofactor of a.
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00:15:25,398 --> 00:15:31,746
And what is the co...
so see, oh no...
211
00:15:31,781 --> 00:15:33,920
I'll then go along the second row.
212
00:15:33,955 --> 00:15:41,022
Ok, so, see which, if I take
now I've got a singular matrix here
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00:15:42,225 --> 00:15:46,499
And I believe that,
when I do this multiplication
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what I'm doing is using the
cofactor formula for the determinant.
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00:15:52,784 --> 00:15:54,615
And then though I'm
going to get a zero
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00:15:54,650 --> 00:15:56,077
Let me try this again
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00:15:56,112 --> 00:15:59,453
So the cofactor formula for
the determinant says
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I should take a times its cofactor
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00:16:04,375 --> 00:16:10,822
which is this b plus b times this cofactor
which is minus a.
220
00:16:10,857 --> 00:16:14,711
Ok, that's what we're doing.
221
00:16:14,746 --> 00:16:16,687
Apart from the sign here
222
00:16:17,099 --> 00:16:23,756
Oh, yea, so it's, it might be
a minus multiplying everything
223
00:16:23,791 --> 00:16:27,746
So if I take this determinant,
the determinant of this
224
00:16:27,781 --> 00:16:30,815
The determinant of As is...
225
00:16:30,850 --> 00:16:34,984
A times its cofactor
which is b
226
00:16:35,019 --> 00:16:39,894
plus the second guy times
this cofactor which is minus a.
227
00:16:40,904 --> 00:16:43,691
And of course I get the answer is zero
228
00:16:43,726 --> 00:16:50,432
and this is exactly what's happening
in that row times this wrong column.
229
00:16:53,461 --> 00:16:57,789
Ok, that's the 2*2 picture
and it's just the same here.
230
00:16:58,213 --> 00:17:03,347
That the reason I get a zero
up in there is...
231
00:17:03,382 --> 00:17:05,089
The reason I get a zero is
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00:17:05,124 --> 00:17:14,842
that when I multiply the first row of a
and the last row of the cofactor matrices
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00:17:14,877 --> 00:17:15,923
It exists
234
00:17:15,958 --> 00:17:21,887
I'm taking this screwed up matrix
that has first and last row identical.
235
00:17:23,965 --> 00:17:28,241
The book prints this down
more specifically
236
00:17:28,241 --> 00:17:31,606
more carefully than
I can do in the lecture
237
00:17:31,058 --> 00:17:32,923
I hope you are seeing the point.
238
00:17:32,958 --> 00:17:35,411
That this is a identity.
239
00:17:35,802 --> 00:17:38,336
That, it is a beautiful identity
240
00:17:38,371 --> 00:17:41,278
and it tells us what the inverse
of a matrix is
241
00:17:42,490 --> 00:17:45,766
So it gives us the inverse,
the formula for the inverse.
242
00:17:45,801 --> 00:17:51,967
Ok, so, that's the first goal of
my lecture was to find this formula.
243
00:17:51,968 --> 00:17:52,968
It's done
244
00:17:53,551 --> 00:17:54,780
Ok
245
00:17:55,644 --> 00:17:57,944
and of course I could inverse now
246
00:17:58,724 --> 00:18:04,875
I can, I sort of, like I can see what,
I can answer questions like this
247
00:18:04,910 --> 00:18:06,620
suppose I have a matrix
248
00:18:08,512 --> 00:18:11,391
and let me remove the 1,1 entry
249
00:18:12,385 --> 00:18:14,141
what happens to the inverse?
250
00:18:14,997 --> 00:18:17,760
Just, just think about that question.
251
00:18:17,795 --> 00:18:19,513
Suppose I have some matrix
252
00:18:19,548 --> 00:18:23,264
I just write down some nice and
nonsingular matrix that's got an inverse
253
00:18:23,299 --> 00:18:26,654
and then I remove the 1,1 entry
a little bit
254
00:18:27,320 --> 00:18:32,513
I add one to it, for example,
what happens to the inverse matrix?
255
00:18:32,548 --> 00:18:34,969
Well, this formula should tell me.
256
00:18:35,004 --> 00:18:38,282
I have to look to see what happens
to the determinant
257
00:18:38,317 --> 00:18:45,209
and what happens to all the cofactors
and the picture is all there.
258
00:18:46,098 --> 00:18:47,233
It's all there
259
00:18:47,268 --> 00:18:53,194
we can really understand how the
inverse changes when the matrix changes
260
00:18:53,229 --> 00:18:59,359
Ok, now my second application is to...
261
00:18:59,768 --> 00:19:02,617
Let me put that over here,
is to Ax equal b
262
00:19:05,306 --> 00:19:10,177
Well, of course
the solution is A inverse b.
263
00:19:11,350 --> 00:19:13,761
But now I have a
formula for A inverse.
264
00:19:13,796 --> 00:19:17,164
A inverse is one over the determinant
265
00:19:19,313 --> 00:19:22,753
times this C transpose times b.
266
00:19:24,162 --> 00:19:26,255
I now know what A inverse is.
267
00:19:26,772 --> 00:19:28,470
So now I just have to say
268
00:19:28,505 --> 00:19:35,688
what do I've got here is there any way
to make this formula, this answer
269
00:19:36,195 --> 00:19:38,527
which is, the one and only answer.
270
00:19:38,562 --> 00:19:40,902
It's the very same answer we've got
271
00:19:41,309 --> 00:19:44,479
on the first day of
the class by elimination
272
00:19:45,018 --> 00:19:47,890
Now I've got, I've got a
formula for the answer.
273
00:19:49,562 --> 00:19:53,627
Can I play with it further to
see what's going on?
274
00:19:54,051 --> 00:20:07,529
And this cramer's rule is exactly,
that a way of looking at this formula
275
00:20:08,921 --> 00:20:11,764
Ok, so this is a formula for X.
276
00:20:13,924 --> 00:20:15,259
Here's my formula.
277
00:20:15,867 --> 00:20:19,066
Well, of course, the first thing
I see from the formula is
278
00:20:19,101 --> 00:20:22,555
that the answer x always
has that in the determinant.
279
00:20:23,095 --> 00:20:26,994
I'm not surprised that it's
the division by the determinant.
280
00:20:27,794 --> 00:20:31,644
But then I have to say a little more
carefully what's going on up here.
281
00:20:31,679 --> 00:20:33,948
And let me tell you what
CRAMER'S RULE is.
282
00:20:33,983 --> 00:20:37,024
Let me take x1, the first component.
283
00:20:38,164 --> 00:20:40,324
So this is the first component
of the answer
284
00:20:40,359 --> 00:20:42,289
then there will be a
second component.
285
00:20:42,856 --> 00:20:44,906
And all the other components.
286
00:20:45,353 --> 00:20:48,891
Can I take just the first component
of this formula?
287
00:20:48,926 --> 00:20:53,115
Well, I certainly have the
determinant of a down under
288
00:20:54,444 --> 00:20:57,869
and what the heck is the first
289
00:20:58,649 --> 00:21:02,048
So what do I get in C transpose b?
290
00:21:02,741 --> 00:21:06,030
What's the first entry of
C transpose b?
291
00:21:06,065 --> 00:21:07,800
That's what I've asked myself.
292
00:21:08,249 --> 00:21:13,026
Well, what's the first
entry of C transpose b?
293
00:21:13,061 --> 00:21:19,732
Ok, er, this b is, let me,
let me tell you what it is
294
00:21:20,514 --> 00:21:27,756
Somehow, I'm multiplying cofactors
by entries of b, right?
295
00:21:27,791 --> 00:21:29,312
In this product.
296
00:21:29,347 --> 00:21:32,454
Cofactors from the
matrix times entries of b.
297
00:21:32,879 --> 00:21:36,274
So any time I'm multiplying cofactors
by numbers
298
00:21:36,309 --> 00:21:38,777
I think I'm getting the
determinants of something.
299
00:21:39,531 --> 00:21:41,916
And let me call that something B1.
300
00:21:43,243 --> 00:21:45,050
So this is a matrix
301
00:21:45,085 --> 00:21:49,886
this is the matrix whose determinant
is coming out of that.
302
00:21:49,921 --> 00:21:51,585
We'll see what it is.
303
00:21:51,620 --> 00:21:56,313
X2 will be the determinant
of some other matrix B2
304
00:21:56,348 --> 00:21:59,365
also divided by determinant of a
305
00:22:00,067 --> 00:22:03,742
so now I just, Cramer just
had a good idea
306
00:22:03,777 --> 00:22:06,552
he realized what that matrix was
307
00:22:06,587 --> 00:22:11,171
what these B1 and B2 and B3
and so on, matrices were.
308
00:22:11,206 --> 00:22:13,502
Let me write them
on the board underneath.
309
00:22:17,344 --> 00:22:19,764
Ok, so what is this B1?
310
00:22:20,695 --> 00:22:27,042
This B1 is the matrix
that has b in its first column
311
00:22:27,077 --> 00:22:30,103
and otherwise the rest of it is A.
312
00:22:30,573 --> 00:22:39,993
So otherwise it has,
this n - 1 columns of A.
313
00:22:41,217 --> 00:22:44,943
It's the matrix with...
314
00:22:44,978 --> 00:22:56,283
it's just the matrix A with column one
replaced by the right hand side
315
00:22:57,446 --> 00:22:59,444
By the right hand side, b.
316
00:23:00,969 --> 00:23:06,735
Because, somehow when I take
the determinant of this guy
317
00:23:07,416 --> 00:23:11,070
it's giving me this
matrix multiplication.
318
00:23:11,688 --> 00:23:13,557
Well, how could that be?
319
00:23:15,990 --> 00:23:20,445
Well, how, what, so what's the
determinant formula I'll use here?
320
00:23:20,913 --> 00:23:22,731
I'll use cofactors, of course.
321
00:23:24,378 --> 00:23:28,040
And now that I'll use cofactors
down column one.
322
00:23:28,075 --> 00:23:30,645
So when I use cofactors
down column one
323
00:23:30,680 --> 00:23:34,430
I'm taking the first entry of b
times what?
324
00:23:35,559 --> 00:23:39,367
Times the cofactor C11.
325
00:23:39,742 --> 00:23:40,565
Do you see that?
326
00:23:41,301 --> 00:23:43,590
When I use cofactors here
327
00:23:43,625 --> 00:23:49,722
I take the first entry here, b1, let's
call it, times the cofactors there.
328
00:23:49,757 --> 00:23:51,482
But what's that cofactor...
329
00:23:51,517 --> 00:23:55,530
My little hand waving indicates that
330
00:23:55,565 --> 00:23:58,776
the matrix is one size
smaller than the cofactor.
331
00:23:58,811 --> 00:24:01,627
And it's exactly c11.
332
00:24:01,662 --> 00:24:03,560
Well that's just what we wanted.
333
00:24:03,595 --> 00:24:08,751
This first entry is c11 times b1
334
00:24:09,475 --> 00:24:17,580
and then the next entry is whatever,
is c21 times b2 and so on
335
00:24:17,615 --> 00:24:19,814
and sure enough, if I look here
336
00:24:19,849 --> 00:24:22,589
when I do the cofactor expansion
337
00:24:22,624 --> 00:24:25,916
b2 is getting multiplied by
the right things and so on.
338
00:24:28,346 --> 00:24:30,266
So there is cramer's rule.
339
00:24:30,301 --> 00:24:35,772
And the book gives another
kind of cute proof without...
340
00:24:36,553 --> 00:24:40,085
without building so much
on the cofactors
341
00:24:40,120 --> 00:24:43,614
but here we've got cofactors
so I thought I just give you this proof
342
00:24:43,649 --> 00:24:47,932
So what is B, in general,
what is Bj?
343
00:24:48,294 --> 00:25:02,206
This is the, this is A with column j
replaced by, replaced by b.
344
00:25:05,446 --> 00:25:10,882
So that's the determinant
of that matrix
345
00:25:10,917 --> 00:25:14,952
divided by that determinant
of A to get exchanged.
346
00:25:14,987 --> 00:25:19,160
So, x, let me change
this general formula, Xj
347
00:25:19,195 --> 00:25:25,641
the jth one is the determinant
of Bj divided by the determinant of A
348
00:25:25,676 --> 00:25:27,872
And now we have said what Bj is.
349
00:25:33,340 --> 00:25:35,871
Well, so Cramer found the rule
350
00:25:36,711 --> 00:25:42,247
And we could ask him,
ok, great, good work Cramer
351
00:25:42,282 --> 00:25:46,267
but is your rule any
good in practice?
352
00:25:47,452 --> 00:25:53,650
So he said, well, you couldn't
ask about a rule of mine, right?
353
00:25:53,685 --> 00:26:00,214
Because all you have to do is
find the determinant of a and b
354
00:26:00,214 --> 00:26:02,528
other determinant
355
00:26:02,563 --> 00:26:04,312
so I guess, I'll just, he just said
356
00:26:04,347 --> 00:26:10,193
Well, all you have to do is find
n + 1 determinant that end b and a
357
00:26:13,822 --> 00:26:15,663
And actually
358
00:26:18,839 --> 00:26:20,840
I remember reading, there was a book
359
00:26:20,875 --> 00:26:23,811
a popular book that kid's
interested in, in math
360
00:26:24,223 --> 00:26:27,452
I read when I was a kid
interested in math
361
00:26:27,487 --> 00:26:32,933
called mathematics for the millions
or something by Grany Bill.
362
00:26:32,968 --> 00:26:36,940
And it has a little page
about linear algebra.
363
00:26:37,695 --> 00:26:39,973
And it said
364
00:26:40,253 --> 00:26:44,048
so it explained the elimination
in a very complicated way.
365
00:26:44,504 --> 00:26:46,338
I certainly didn't understand it.
366
00:26:46,373 --> 00:26:49,790
And it made it, you know
it sort of said
367
00:26:50,737 --> 00:26:54,257
Well, there is a formula
for elimination
368
00:26:54,751 --> 00:26:57,732
but look at this great formula
Cramer's Rule.
369
00:26:58,442 --> 00:27:02,154
So it certainly said Cramer's Rule
was a way to go
370
00:27:03,737 --> 00:27:08,675
But actually, Cramer's Rule is
the disastrous way to go
371
00:27:08,918 --> 00:27:15,091
Because to compute this determinant
takes like, possibly forever.
372
00:27:15,126 --> 00:27:22,393
So actually I now think of that book title
being mathematics for the millionaire
373
00:27:22,428 --> 00:27:28,624
because you'd have to be able to
pay for a hopeless long calculation
374
00:27:28,659 --> 00:27:33,331
where the elimination
of course to do is in an instant.
375
00:27:33,744 --> 00:27:38,311
But having a formula allows you to...
376
00:27:38,346 --> 00:27:43,474
whether, you know, it allows you to
do algebra instead of algorithm.
377
00:27:43,509 --> 00:27:48,833
So there is some value in
the Cramer's rule of formula for x
378
00:27:48,868 --> 00:27:53,990
and in the explicit formula
for A inverse.
379
00:27:54,802 --> 00:27:57,167
It's...they are nice formulas
380
00:27:57,202 --> 00:28:00,533
but I just don't want you to use them
381
00:28:00,880 --> 00:28:02,405
That's, that's what I come to.
382
00:28:02,440 --> 00:28:06,511
If you had to use...
I would never, never do that.
383
00:28:06,546 --> 00:28:08,817
I mean, if must use an elimination.
384
00:28:08,852 --> 00:28:09,625
Ok
385
00:28:10,831 --> 00:28:17,960
Now I'm ready for No.3 in today's list
of amazing connections
386
00:28:18,391 --> 00:28:20,395
coming through the determinant
387
00:28:21,343 --> 00:28:26,176
and that No.3 is the fact
that the determinant gives a volume
388
00:28:26,946 --> 00:28:33,237
Ok, So that's my final topic for...
among these
389
00:28:34,276 --> 00:28:36,397
it's my No.3 application.
390
00:28:36,872 --> 00:28:41,046
That the determinant is actually
equal to the volume of something.
391
00:28:41,081 --> 00:28:45,849
Can I use this little space to
consider a special case
392
00:28:46,710 --> 00:28:51,396
and then I'll use a far board
to think about the general rule.
393
00:28:51,431 --> 00:28:55,519
So what am I gonna prove, or claim?
394
00:28:55,554 --> 00:29:05,260
I claim that the determinant of
the matrix is the volume of a box
395
00:29:06,437 --> 00:29:11,307
Ok, and you say, which box?
Where now, Ok.
396
00:29:12,218 --> 00:29:13,210
So...
397
00:29:13,991 --> 00:29:18,859
Let's see, I'm in,
so we say we are in say, 3*3?
398
00:29:19,643 --> 00:29:24,568
so we suppose, let's take 3*3,
so we can really...
399
00:29:24,603 --> 00:29:29,756
we are talking about boxes and
three dimensions and 3*3 matrices
400
00:29:30,119 --> 00:29:34,219
And all I'll do, you could
guess what the box is.
401
00:29:34,254 --> 00:29:37,738
Here's, here's the three dimensions
402
00:29:38,131 --> 00:29:42,343
Ok, and now I take the
first row of the matrix.
403
00:29:43,670 --> 00:29:50,677
A11, a22, sorry, a11, a12, a13,
that row is a vector.
404
00:29:51,324 --> 00:29:53,852
It goes to some points
405
00:29:53,887 --> 00:29:58,698
that point will be and that edge going
to it will be the edge of the box
406
00:29:58,733 --> 00:30:01,273
and that point will
be a corner of the box.
407
00:30:01,308 --> 00:30:03,792
So here's (0, 0, 0)
of course
408
00:30:04,584 --> 00:30:07,474
and here's the first row
of the matrix.
409
00:30:07,898 --> 00:30:12,907
(a11, a12, a13).
410
00:30:14,039 --> 00:30:16,794
So that's one edge of the box.
411
00:30:18,001 --> 00:30:24,348
Another edge of the box is two
the second row of the matrix.
412
00:30:24,383 --> 00:30:27,897
Row two,
can I just call that row two?
413
00:30:27,932 --> 00:30:31,917
And the third row of the box
will be two...
414
00:30:31,952 --> 00:30:36,668
The third row, the third edge of the
box will be given by row three
415
00:30:37,070 --> 00:30:39,240
So, so there's row three.
416
00:30:40,070 --> 00:30:44,334
That, that's sort of the corner
the corner of the box.
417
00:30:45,279 --> 00:30:51,271
(a31, a32, a33).
418
00:30:52,039 --> 00:30:54,030
So I've got that edge of the box
419
00:30:54,065 --> 00:30:57,791
that edge of the box, that edge
of the box, and that's all I need
420
00:30:57,826 --> 00:30:59,475
And now I'll just finish all the box.
421
00:30:59,510 --> 00:31:02,648
Right? I'll just make....
sorry
422
00:31:02,683 --> 00:31:12,616
box, the proper word is paroling piper
but for obvious reasons, I wrote box
423
00:31:12,651 --> 00:31:17,831
Ok, so, er, so there's the,
there's the bottom of the box.
424
00:31:18,232 --> 00:31:23,558
There's the four edge side of the box.
425
00:31:23,593 --> 00:31:25,274
There's the top of the box.
426
00:31:27,608 --> 00:31:34,484
Right? It's the box that has these three
edges and then completed the two of...
427
00:31:34,519 --> 00:31:39,898
the two of, you know,
each side is a parallel diagram.
428
00:31:41,542 --> 00:31:47,315
And it's that box, whose volume
is given by the determinant.
429
00:31:49,384 --> 00:31:55,317
That's, now it's possible that
the determinant is negative.
430
00:31:56,917 --> 00:31:59,856
So we'll have to just say
what's going on in that case.
431
00:31:59,891 --> 00:32:05,654
If the determinant is negative,
then the volume...
432
00:32:05,689 --> 00:32:08,058
we should take the absolute
value really
433
00:32:08,093 --> 00:32:11,023
so the volume, if we think of
the volume is positive
434
00:32:11,058 --> 00:32:14,122
we should take the absolute
value of the determinant.
435
00:32:15,180 --> 00:32:18,582
What is the sign, what's
the sign of the determinant
436
00:32:18,617 --> 00:32:20,149
It always must tell us something.
437
00:32:20,566 --> 00:32:24,187
And somehow it will tell us
whether these three...
438
00:32:24,222 --> 00:32:27,341
whether it's a right handed box,
or it's a left handed box.
439
00:32:27,376 --> 00:32:32,395
If we reverse the two of the edges
440
00:32:34,274 --> 00:32:37,599
we would go between a right
handed box and a left handed box
441
00:32:37,634 --> 00:32:42,604
We will not change the value
but we will change the sickly.
442
00:32:42,639 --> 00:32:43,600
Here order
443
00:32:44,418 --> 00:32:46,393
So that, I wouldn't
worry about that.
444
00:32:46,784 --> 00:32:49,708
And this, one special case is what?
445
00:32:50,824 --> 00:32:53,221
A equal identity matrix.
446
00:32:53,967 --> 00:32:55,806
So let's take that special case.
447
00:32:55,841 --> 00:32:59,445
A equal identity matrix.
448
00:32:59,480 --> 00:33:00,971
It's the formula
449
00:33:01,006 --> 00:33:05,800
the determinant of identity matrix,
did that equal the volume of the box?
450
00:33:08,313 --> 00:33:09,701
But what is the box?
451
00:33:10,512 --> 00:33:13,855
What's the box if A is
the identity matrix
452
00:33:13,890 --> 00:33:19,620
then these three rows are the three
coordinators vectors, and the box is?
453
00:33:21,767 --> 00:33:22,708
It's a cube.
454
00:33:22,743 --> 00:33:24,445
It's the unit cube.
455
00:33:25,250 --> 00:33:29,386
So if a is the identity matrix,
of course our formula is right.
456
00:33:30,013 --> 00:33:33,243
Well, actually that
proves property one.
457
00:33:33,953 --> 00:33:37,543
That the volume has property one,
actually we could, we could...
458
00:33:37,864 --> 00:33:41,047
We could get this thing
if we can show that
459
00:33:41,082 --> 00:33:45,893
the box volume has the same three
properties that define the determinant
460
00:33:46,598 --> 00:33:48,490
then it must be the determinant.
461
00:33:50,868 --> 00:33:54,545
So that's like the elegant
way to prove it.
462
00:33:55,274 --> 00:33:58,915
The proof is an amazing fact that
the determinant equals a volume
463
00:33:58,950 --> 00:34:01,782
first we'll check the identity matrix.
464
00:34:02,541 --> 00:34:06,132
That's fine, the box is a cube
465
00:34:06,167 --> 00:34:08,981
and its volume is one and
then the determinant is one
466
00:34:09,016 --> 00:34:10,727
and one agrees with one.
467
00:34:11,512 --> 00:34:16,697
Now let me take one, let me go up
one level to an orthogonal Matrix
468
00:34:17,003 --> 00:34:22,102
Because I'd like to take this chance to
bring in the previous chapter
469
00:34:22,728 --> 00:34:25,563
Suppose I have an orthogonal matrix
470
00:34:25,598 --> 00:34:28,839
what does that mean that
I would call those thing cube?
471
00:34:29,598 --> 00:34:31,949
What was the point of,
suppose I have...
472
00:34:31,984 --> 00:34:34,656
Suppose instead of the identity matrix
473
00:34:34,691 --> 00:34:41,742
I'm now going to take A equal Q,
an orthogonal matrix.
474
00:34:47,105 --> 00:34:49,675
What's that, what was Q then?
475
00:34:50,677 --> 00:34:55,420
That's a matrix whose columns
were orthogonal, right?
476
00:34:56,364 --> 00:35:00,818
Those columns were unit vectors,
perpendicular unit vectors.
477
00:35:03,801 --> 00:35:05,851
So what kind of box
do we have got now?
478
00:35:07,609 --> 00:35:11,647
What a kind of box comes from the
rows or the columns I don't mind
479
00:35:11,682 --> 00:35:13,163
Because it's the determinant
480
00:35:13,198 --> 00:35:17,033
it's the determinant transpose
and I'll never worry about that.
481
00:35:17,722 --> 00:35:19,121
What kind of box,
482
00:35:19,156 --> 00:35:23,711
what shape of box shall we've got
if the matrix is an orthogonal matrix?
483
00:35:24,429 --> 00:35:28,506
It's another cube,
it's a cube again.
484
00:35:29,097 --> 00:35:32,482
How is it different from
the identity cube?
485
00:35:34,834 --> 00:35:36,685
It's just rotated.
486
00:35:37,526 --> 00:35:43,298
It's just the orthogonal matrix cube
doesn't have to be the identity matrix
487
00:35:43,333 --> 00:35:47,107
It's just the unit cube
but turned in space.
488
00:35:47,629 --> 00:35:52,704
So sure enough, it's the unit
cube and its volume is one
489
00:35:52,739 --> 00:35:56,441
Now is the determinant one?
490
00:35:57,801 --> 00:36:00,096
What's the determinant of Q?
491
00:36:00,131 --> 00:36:03,994
We believe the determinant Q
better be one or minus one.
492
00:36:04,433 --> 00:36:07,146
So that our formula is
checked out in that.
493
00:36:07,530 --> 00:36:11,303
If we can't check it in easy cases
where we got a cube
494
00:36:11,659 --> 00:36:14,440
We are not gonna get it
in the general case.
495
00:36:15,520 --> 00:36:20,889
So why is the determinant of Q
equal one or minus one?
496
00:36:22,766 --> 00:36:24,416
What do we know about Q?
497
00:36:24,451 --> 00:36:28,763
What's the one matrix specimen
of the property of Q?
498
00:36:29,771 --> 00:36:35,881
The matrix with orthogonal columns
has satisfied a certain equation.
499
00:36:36,579 --> 00:36:37,839
What is that?
500
00:36:37,874 --> 00:36:41,198
It's...
if we have this orthogonal matrix
501
00:36:41,714 --> 00:36:49,267
then the fact, the other way to say
what the properties are
502
00:36:49,302 --> 00:36:56,291
It is Q transpose Q equals I.
Right?
503
00:36:57,152 --> 00:37:02,126
That's what, those were the
matrices that get the name Q.
504
00:37:02,161 --> 00:37:05,478
The matrices that Q transpose Q is I.
505
00:37:06,218 --> 00:37:13,679
Ok, now from that, tell me why is the
determinant one or minus one.
506
00:37:14,361 --> 00:37:19,708
How do I, out of this fact
It's being a homework problem.
507
00:37:22,032 --> 00:37:26,872
It's there in the list of the exercises
in the book and let's just do it.
508
00:37:27,643 --> 00:37:28,880
How do I get...
509
00:37:28,915 --> 00:37:34,775
how do I discover that the determinant
of Q is one or maybe minus one?
510
00:37:36,242 --> 00:37:39,102
I take the determinant on
both side, everybody says.
511
00:37:39,137 --> 00:37:44,456
So I won't, I take the determinant is
a both side, so on the right hand side
512
00:37:44,491 --> 00:37:48,619
So when I take the determinant
on both side, let me just do it.
513
00:37:51,102 --> 00:37:53,525
Take the determinant of...
take determinants
514
00:37:56,072 --> 00:37:58,409
determinant of the identity is one.
515
00:37:58,444 --> 00:38:01,038
What's the determinant of
that product?
516
00:38:03,677 --> 00:38:06,258
Rule nine is paying off now.
517
00:38:07,229 --> 00:38:12,188
The determinant of the product is
the determinant of this guy
518
00:38:12,787 --> 00:38:16,158
maybe I will put it, I'll use that
symbol for the determinant
519
00:38:16,193 --> 00:38:20,134
It's the determinant of that guy
times the determinant of the other guy
520
00:38:21,907 --> 00:38:25,108
And then what's the
determinant of Q transpose?
521
00:38:26,260 --> 00:38:29,403
It's the same as determinatnt Q
routine pays off.
522
00:38:29,438 --> 00:38:32,220
So this is just,
this thing square.
523
00:38:33,276 --> 00:38:35,399
So that determinant of square is one
524
00:38:35,434 --> 00:38:39,895
and sure enough,
it's one or minus one.
525
00:38:40,243 --> 00:38:41,077
Great.
526
00:38:43,272 --> 00:38:46,450
So in these special cases
that have cubes
527
00:38:47,474 --> 00:38:52,741
we really do have
determinants equal volume.
528
00:38:53,690 --> 00:38:58,756
Now can I just push that
to non-cubes?
529
00:39:00,087 --> 00:39:05,541
Let me push it
first to rectangles.
530
00:39:05,576 --> 00:39:07,809
Rectangular boxes.
531
00:39:08,636 --> 00:39:12,744
Well, I'm just multiplying the edges
the edges are...
532
00:39:12,779 --> 00:39:15,228
Let me keep all the 90 degree angles
533
00:39:15,263 --> 00:39:18,034
Coz those, that makes my life easy.
534
00:39:18,509 --> 00:39:21,243
And just stretch the...
stretch the edges.
535
00:39:21,985 --> 00:39:24,682
Suppose I stretch that first edge
536
00:39:25,246 --> 00:39:28,271
Suppose that's the first
edge I double
537
00:39:28,306 --> 00:39:30,534
Suppose I doubled that first edge
538
00:39:32,838 --> 00:39:34,774
keeping the other edges the same
539
00:39:36,064 --> 00:39:37,587
What happens to the volume?
540
00:39:39,658 --> 00:39:41,028
It doubles, right?
541
00:39:41,590 --> 00:39:45,216
We know that the volume of
a cube doubles in fact
542
00:39:45,251 --> 00:39:48,484
Because we know that the new cube
would sit right on top of...
543
00:39:48,519 --> 00:39:52,480
I mean the new, the added
cube would sit right on, would fit...
544
00:39:52,515 --> 00:39:57,892
Probably algebra would say coagula into
the right in the other one, we have two
545
00:39:57,927 --> 00:40:01,370
If two identical cubes,
total volume is now two.
546
00:40:03,649 --> 00:40:07,622
Ok, what, so if I double an edge
the volume doubles.
547
00:40:07,657 --> 00:40:09,345
What happens to the determinant?
548
00:40:12,157 --> 00:40:16,713
If I double the first row
of the matrix
549
00:40:19,491 --> 00:40:22,516
What's the effect on the determinant?
550
00:40:23,081 --> 00:40:25,974
It also doubles, right?
551
00:40:26,367 --> 00:40:30,903
And that was the rule number 3a.
552
00:40:31,749 --> 00:40:38,839
Remember rule 3a was, that if I could
if I had a factor in row one, p
553
00:40:39,295 --> 00:40:40,827
I could subtract it out.
554
00:40:41,972 --> 00:40:46,185
So if I have a factor two
in that row one
555
00:40:46,220 --> 00:40:48,314
I can fact without the determinant
556
00:40:48,349 --> 00:40:52,859
and agrees with the volume
of the box satisfy fact two
557
00:40:54,081 --> 00:40:57,789
so volumes satisfy
this property 3a.
558
00:40:58,992 --> 00:41:05,198
And now I'm really, I'm close, but I
get to the very end of the truth
559
00:41:05,233 --> 00:41:07,911
I have to get away from right angle
560
00:41:07,946 --> 00:41:14,285
I have to allow the possibility of
other angles
561
00:41:15,562 --> 00:41:18,371
And or, I'm saying the same thing
562
00:41:18,406 --> 00:41:23,323
I have to check the volume
also satisfies 3b.
563
00:41:24,015 --> 00:41:28,735
So, can I, can I just,
so this is the end of truth.
564
00:41:28,770 --> 00:41:39,398
The determinant of A equals volume
of box, and where am I write now?
565
00:41:39,433 --> 00:41:45,491
This volume has property,
property one, no problem.
566
00:41:46,307 --> 00:41:52,052
It's the box of a cube, everything is,
it's the box with the unit cube.
567
00:41:52,087 --> 00:41:53,291
Its volume is one.
568
00:41:53,853 --> 00:41:59,617
Property two was,
if I reverse two rows
569
00:42:00,102 --> 00:42:02,540
But that doesn't change the box
570
00:42:03,218 --> 00:42:06,517
and it doesn't change the absolute
value, so no problem there.
571
00:42:06,552 --> 00:42:12,169
Property 3a was, if I,
you remember what 3a was?
572
00:42:12,204 --> 00:42:15,228
So property one was
about the identity matrix
573
00:42:15,689 --> 00:42:19,561
Property 2 was about a plus or
minus sign that I don't care about
574
00:42:19,596 --> 00:42:23,318
Property 3a was a factor T in a row
575
00:42:23,716 --> 00:42:27,452
but now I've got property 3b
to deal with.
576
00:42:28,936 --> 00:42:35,972
What's property 3b, this is a great
way to review these properties.
577
00:42:36,007 --> 00:42:39,378
So 3b, property 3b says
578
00:42:39,413 --> 00:42:41,816
Let's do, let's do 2*2.
579
00:42:42,306 --> 00:42:48,618
So said it, if I have a plus a prime,
b plus b prime, c, d.
580
00:42:50,365 --> 00:42:52,235
That this equals what?
581
00:42:53,003 --> 00:42:58,209
So this is property 3b, this is
the linearity in row one by itself
582
00:42:59,417 --> 00:43:01,449
So c, d is staying the same
583
00:43:01,484 --> 00:43:08,102
and I can split this into a, b
and a prime, b prime.
584
00:43:11,421 --> 00:43:13,540
That's property 3b.
585
00:43:14,580 --> 00:43:16,765
This is the 2*2 case.
586
00:43:18,542 --> 00:43:20,727
And what am I...
587
00:43:20,762 --> 00:43:23,869
I wanna now to show
that the volume
588
00:43:25,432 --> 00:43:31,912
which 2*2, that means area,
has this, has this property.
589
00:43:34,134 --> 00:43:36,117
Let me just emphasize...
590
00:43:36,152 --> 00:43:38,227
We've got, we are getting...
591
00:43:38,262 --> 00:43:42,127
this is a formula then
for the area of a parallelogram.
592
00:43:42,865 --> 00:43:45,726
The area of this parallelogram,
can I just draw it?
593
00:43:45,761 --> 00:43:49,930
Ok, here's, here's the
parallelogram, I've the row a, b.
594
00:43:50,777 --> 00:43:54,058
That's the first row,
that's the point (a,b).
595
00:43:54,888 --> 00:43:59,020
And I tack on (c,d).
596
00:44:00,051 --> 00:44:02,362
(c,d) coming out of here.
597
00:44:03,310 --> 00:44:05,388
And I complete the parallelogram.
598
00:44:07,554 --> 00:44:13,453
So this is c, well, I'd better
make it look, right,
599
00:44:13,488 --> 00:44:17,207
it's really, it's this one
that has the coordinator (c,d).
600
00:44:17,242 --> 00:44:21,697
And this has the coordinator,
well, whatever the sum is.
601
00:44:22,631 --> 00:44:24,886
And of course, starting at (0,0).
602
00:44:27,906 --> 00:44:31,906
So we all know,
this is a plus c, b plus d.
603
00:44:34,514 --> 00:44:35,806
Rather than...
604
00:44:37,028 --> 00:44:41,023
I'm pausing on that proof for a minute
just going back to our formula.
605
00:44:42,865 --> 00:44:44,848
Because I want you to see that,
606
00:44:44,883 --> 00:44:48,535
unlike Cramer's rule,
that I was't, that impressed by.
607
00:44:49,259 --> 00:44:53,565
I'm very impressed by this formula for
the area of a parallelogram.
608
00:44:54,311 --> 00:44:55,559
And what's our formula?
609
00:44:56,337 --> 00:44:59,113
What, what's the area
of that parallelogram?
610
00:45:00,082 --> 00:45:05,569
If I asked you that last year
611
00:45:06,444 --> 00:45:07,838
You would have said: ok
612
00:45:07,873 --> 00:45:12,363
the area of a parallelogram is
the base times the height, right?
613
00:45:13,834 --> 00:45:18,336
So you would have figured out what
this base, so how long that base was?
614
00:45:18,477 --> 00:45:21,904
It's like the square rule
of a square and b square.
615
00:45:21,939 --> 00:45:25,802
And then you would have figured out
how much is this height, wherever it is.
616
00:45:25,837 --> 00:45:27,157
It's horrible.
617
00:45:27,639 --> 00:45:32,917
I mean, we've got square rows.
618
00:45:32,952 --> 00:45:37,922
And in that height, there
would be other revolting stuffs.
619
00:45:37,957 --> 00:45:41,797
But now what's the formula
that we now know for the area?
620
00:45:48,145 --> 00:45:52,056
It's the determinant
of our little matrix.
621
00:45:53,842 --> 00:45:58,878
It's just ad minus bc.
622
00:46:02,819 --> 00:46:07,222
No square rows,
totally remembable
623
00:46:07,257 --> 00:46:08,988
because it's exactly a formular
624
00:46:09,023 --> 00:46:13,703
that we've been studying the whole
for three lectures.
625
00:46:14,854 --> 00:46:20,949
Ok, that's, you know, that's the
most important point I'm making here
626
00:46:20,984 --> 00:46:28,689
If you know the coordinator
of a box of the corners
627
00:46:29,204 --> 00:46:33,327
then you have a great formula
for the volume, area volume
628
00:46:34,543 --> 00:46:39,179
that doesn't involve any lanes or
any angles or any heights.
629
00:46:39,911 --> 00:46:42,817
But just involves the coordinators
that you've got.
630
00:46:43,557 --> 00:46:46,477
And similarly, what's the
area of this triangle?
631
00:46:46,512 --> 00:46:49,449
Suppose that I chop that off,
and say what about...
632
00:46:49,484 --> 00:46:53,104
Coz you might be often more interested
in a triangle instead of a parallelogram
633
00:46:53,139 --> 00:46:55,236
what's the area of the triangle?
634
00:46:57,957 --> 00:46:58,905
Now there again.
635
00:46:58,940 --> 00:47:00,568
Everybody would have said that
636
00:47:00,603 --> 00:47:04,549
the area of that triangle is
half the base times the height.
637
00:47:07,251 --> 00:47:11,800
And in some cases, if you know
the base and height, that's fine.
638
00:47:12,485 --> 00:47:15,911
But here, what we know is
the coordinators of the corners.
639
00:47:15,946 --> 00:47:18,105
We know the vectors.
640
00:47:18,140 --> 00:47:21,345
And so, what's the area
of that triangle?
641
00:47:24,007 --> 00:47:28,371
If I know these, if I
know (a,b), (c,d) and (0,0)
642
00:47:28,406 --> 00:47:30,260
What's the co, what's the area?
643
00:47:30,761 --> 00:47:35,900
It's just half,
so, it's just half of this.
644
00:47:35,935 --> 00:47:42,023
So this is ad minus bc
for the parallelogram
645
00:47:42,800 --> 00:47:50,551
and one half of that, one half
of ad minus bc for the triangle.
646
00:47:53,430 --> 00:47:58,172
So I doesn't, I mean this is the totally
trivial remark to say that divided by 2
647
00:47:58,940 --> 00:48:02,136
But it's just that you more
often see triangles
648
00:48:03,850 --> 00:48:07,748
and you feel, you know
the formula for the area
649
00:48:07,783 --> 00:48:10,632
but the good formula
for the area is this one.
650
00:48:11,292 --> 00:48:12,488
And I'm just going to...
651
00:48:12,814 --> 00:48:16,049
I'm just going to say one more thing
about the area of a triangle
652
00:48:16,084 --> 00:48:21,500
Just coz, just you know, it's so great
to have a good formula for some thing
653
00:48:22,060 --> 00:48:25,468
What if our triangle
did not start at (0,0)
654
00:48:27,058 --> 00:48:28,860
what if our triangle...
655
00:48:28,895 --> 00:48:32,695
what if we had, what if we had...
656
00:48:32,730 --> 00:48:34,727
So I'm coming back to triangles again
657
00:48:38,712 --> 00:48:46,405
But let me, let me put that triangle
somewhere, it's...
658
00:48:46,440 --> 00:48:49,999
I'm saying with triangles, so I'm
just in two-dimension.
659
00:48:50,034 --> 00:49:00,536
But I'm, I allow you to
give me any three corners.
660
00:49:03,683 --> 00:49:07,392
And in those six numbers
must determine the area.
661
00:49:07,817 --> 00:49:09,408
And what's the formula?
662
00:49:10,068 --> 00:49:15,052
The area is going to be
the half of a parallelogram
663
00:49:15,899 --> 00:49:19,993
I mean basically, it's can't
be completely new, right?
664
00:49:20,028 --> 00:49:24,432
We got the area when, we know
the area when this is (0,0).
665
00:49:26,662 --> 00:49:30,122
Now we just want a...slightly
666
00:49:30,157 --> 00:49:32,640
and get the area when all...
667
00:49:32,675 --> 00:49:35,730
so let me write down what it
comes out to be
668
00:49:35,765 --> 00:49:41,779
It turns out that, if you do this
x1, y1 and a 1
669
00:49:41,814 --> 00:49:48,340
x2, y2 and a 1,
x3, y3 and a 1
670
00:49:48,765 --> 00:49:49,713
Dose that work?
671
00:49:50,211 --> 00:49:53,107
That's the determinant symbol
of course
672
00:49:53,886 --> 00:49:54,709
It's just...
673
00:49:55,944 --> 00:49:58,156
If I give you that
the determinant to find
674
00:49:58,191 --> 00:50:01,693
you might subtract this row
from this, it will kill that one.
675
00:50:02,143 --> 00:50:04,975
Subtract this from this,
it will kill that one.
676
00:50:05,458 --> 00:50:11,554
Then you'll have a simple determinant
to do with differences, and with...
677
00:50:12,383 --> 00:50:15,476
This little subtraction what I did
678
00:50:15,511 --> 00:50:21,607
equivalent to moving the triangle
to start the argent.
679
00:50:22,686 --> 00:50:24,361
I did it fast.
680
00:50:24,396 --> 00:50:25,483
Coz time is up.
681
00:50:26,264 --> 00:50:30,302
And I didn't complete
that proof of 3d
682
00:50:31,874 --> 00:50:36,555
I will leave the book has carefully
drawn and figured so why that works
683
00:50:37,235 --> 00:50:40,345
But I hope you saw the main point is
684
00:50:40,380 --> 00:50:45,424
that for area volume,
determinant gives a great formula
685
00:50:45,882 --> 00:50:51,003
Ok, and next lectures are
about eigenvalues
686
00:50:51,038 --> 00:50:54,871
so we're really into the big stuffs
687
00:50:54,906 --> 00:50:56,042
Thanks.
Last Modified 3/30/08 4:41 AM
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