WEBVTT
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OK, this is lecture twenty.
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And this is the final
lecture on determinants.
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And it's about the applications.
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So we worked hard in
the last two lectures
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to get a formula
for the determinant
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and the properties
of the determinant.
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Now to use the determinant and,
and always this determinant
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packs all this information
into a single number.
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And that number can
give us formulas
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for all sorts of, things that
we've been calculating already
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without formulas.
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Now what was A inverse?
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So, so I'm beginning with
the formula for A inverse.
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Two, two by two formula we know,
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right?
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The two by two formula for A
inverse, the inverse of a b c d
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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 three by three
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and n by n.
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And actually maybe you can see
what's going on from this two
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by two case.
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So there's a formula
for the inverse,
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and what did I divide by?
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The determinant.
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So what I'm looking
for is a formula
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where it has one
over the determinant
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and, and you remember why
that makes good sense,
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because then that's perfect as
long as the determinant isn't
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zero, 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 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, my formula for the inverse
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is going to be one
over the determinant
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times a matrix of cofactors.
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So you remember that D?
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What's that the cofactor of?
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Remember cofactors?
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If -- that's the
one one 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 minus b?
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OK.
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Which cofactor is that one?
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Oh, minus 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 the minus sign?
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Because this c was in a two one
position, and two plus one is
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odd.
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So there was a minus went into
the cofactor, and that's it.
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OK.
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I'll write down next
what my formula is.
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Here's the big formula for
the A -- for A inverse.
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It's one over the determinant
of A 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, it
turns -- you'll see,
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I have to, I transpose.
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So this is the matrix
of cofactors, the --
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what I'll just --
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but why don't we just call
it the cofactor matrix.
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So the one one entry of, of
c is the cof- is the one one
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cofactor, the thing that we get
by throwing away row and column
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one.
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It's the d.
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And, because of the
transpose, what I see up here
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is the cofactor of this
guy down here, right?
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That's where the
transpose came 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.
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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 c, and then
I have that minus sign
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again.
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And of course the two
two entry is the cofactor
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of d, and that's this a.
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So there's the formula.
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OK.
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But we got to think why.
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I mean, it worked in
this two by two case,
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but a lot of other formulas
would have worked just as well.
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We, we have to see
why that's true.
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In other words, why is it --
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so this is what I aim to find.
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And, and let's just sort of
look to see what is that telling
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us.
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That tells us that the -- the
expression for A inverse --
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let's look at a three by three.
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Can I just write down
a 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 a
formula -- do I see there?
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I mean, what --
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the determinant is a bunch
of products of three factors,
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right?
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The determinant of this
matrix'll involve a e i,
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and b f times g, and c
times d times h, and minus c
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e g, and so on.
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So things with three
factors go in here.
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Things with how many factors do
things in the cofactor matrix
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have?
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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 one one
entry up here in the inverse
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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,
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that's the cofactor.
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And that's e i minus f h.
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Products of two things.
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Now, I'm just making
the observation
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that the determinant of A
involves products of n entries.
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And the cofactor matrix involves
products of n minus 1 entries.
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And, like, we never
noticed any of this stuff
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when we were
computing the inverse
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by the Gauss-Jordan
method or whatever.
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You remember how we did it?
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We took the matrix A, we
tucked the identity next to it,
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we did elimination till
A became the identity.
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And then that, the identity
suddenly was A inverse.
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Well, that was
great numerically.
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But we never knew what
was going on, basically.
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And now we see
what the formula is
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in terms of letters,
what's the algebra instead
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of the algorithm.
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OK.
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But I have to say why
this is right, right?
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I still -- 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 got it
up there, let me --
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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
its inverse gives the identity.
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So I want, I want to check
that A times this thing,
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A times this -- now I'm going
to write in the inverse --
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gives the identity.
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So I 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.
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That's it, that if this is
true, and it is, then, then I've
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correctly identified A
inverse as C transpose divided
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by the determinant.
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OK.
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But why is this true?
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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 A --
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here, here's A, here's a11 --
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I'm doing this multiplication
-- along to a1n.
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And then down in this last row
will be an an1 along to ann.
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And I'm multiplying by the
cofactor matrix transposed.
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So when I transpose, it'll
be c11 c12 down to c1n.
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Notice usually that
one coming first
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would mean I'm in row
one, but I've transposed,
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so that's, those
are the cofactors.
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This first column are the
cofactors from row one.
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And then the last column 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
one one entry in the product
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is.
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This is like you're seeing
the main point if you just
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tell me one entry.
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What do I get up there
in the one one entry
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when I do this row of
this row from A times
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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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I mean, we're, right,
building exactly
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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 does that what
does that sum up to?
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That's the cofactor formula
for the determinant.
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That's the, this
cofactor formula,
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which I wrote, which
we got last time.
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The determinant of A is,
if I use row one, let,
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let I equal one,
then I have a11 times
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its cofactor, a12 times
its cofactor, and so on.
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And that gives me
the determinant.
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And it worked in
this, in this case.
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This row times this thing
is, sure enough, ad minus bc.
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But this formula
says it always works.
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So up here in this,
in this position,
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I'm getting determinant of A.
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What about in the
two two position?
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Row two times column two
there, what, what is that?
00:11:23.460 --> 00:11:26.640
That's just the cofactors,
that's just row two
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times its cofactors.
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So of course I get
the determinant again.
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And in the last here,
this is the last row
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times its cofactors.
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It's exactly -- you see, we're
realizing that the cofactor
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formula is just this
sum of products,
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so of course we think,
hey, we've got a matrix
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multiplication there.
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And we get determinant of A.
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But there's one more
idea here, right?
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Great.
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What else, what have I not --
so I haven't got that formula
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completely proved yet, because
I've still got to do all
00:12:10.520 --> 00:12:14.130
the off-diagonal stuff,
which I want to be zero,
00:12:14.130 --> 00:12:15.120
right?
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I just want this
to be determinant
00:12:17.560 --> 00:12:22.850
of A times the identity, and
then I'm, I'm a happy person.
00:12:22.850 --> 00:12:24.860
So why should that be?
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Why should it be that one
row times the cofactors
00:12:30.660 --> 00:12:34.920
from a different row, which
become a column because I
00:12:34.920 --> 00:12:39.360
transpose, give zero?
00:12:39.360 --> 00:12:44.210
In other words, the cofactor
formula gives the determinant
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if the row and the, and
the cofactors -- you know,
00:12:48.440 --> 00:12:54.050
if the entries of A and the
cofactors are for the same row.
00:12:54.050 --> 00:12:58.470
But for some reason, if I
take the cofactors from the --
00:12:58.470 --> 00:13:01.920
entries from the first row and
the cofactors from the second
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row, for some reason
I automatically
00:13:04.290 --> 00:13:05.270
get zero.
00:13:05.270 --> 00:13:08.140
And it's sort of
like interesting
00:13:08.140 --> 00:13:10.540
to say, why does that happen?
00:13:10.540 --> 00:13:14.620
And can I just check that --
of course, we know it happens,
00:13:14.620 --> 00:13:16.410
in this case.
00:13:16.410 --> 00:13:19.060
Here are the
numbers from row one
00:13:19.060 --> 00:13:24.600
and here are the cofactors
from row two, right?
00:13:24.600 --> 00:13:26.820
Those are the
numbers in row one.
00:13:26.820 --> 00:13:29.510
And th- these are the
cofactors from row two,
00:13:29.510 --> 00:13:32.110
because the cofactor
of c is minus b
00:13:32.110 --> 00:13:33.530
and the cofactor of d is
00:13:33.530 --> 00:13:34.030
a.
00:13:34.030 --> 00:13:40.380
And sure enough, that row
times this column gives --
00:13:40.380 --> 00:13:41.840
please say it.
00:13:41.840 --> 00:13:43.370
Zero, right.
00:13:43.370 --> 00:13:43.870
OK.
00:13:43.870 --> 00:13:47.330
So now how come?
00:13:47.330 --> 00:13:48.250
How come?
00:13:48.250 --> 00:13:51.190
Can we even see it in
this two by two case?
00:13:51.190 --> 00:13:54.460
Why did -- well, I mean,
I guess we, you know,
00:13:54.460 --> 00:13:57.352
in one way we certainly do see
it, because it's right here.
00:13:57.352 --> 00:13:59.310
I mean, do we just do
it, and then we get zero.
00:13:59.310 --> 00:14:01.870
But we want to
think of some reason
00:14:01.870 --> 00:14:07.280
why the answer's zero, some
reason that we can use in the n
00:14:07.280 --> 00:14:08.820
by n case.
00:14:08.820 --> 00:14:10.590
So let -- here,
here is my thinking.
00:14:13.450 --> 00:14:16.310
We must be, if we're
getting the answer's zero,
00:14:16.310 --> 00:14:20.330
we suspect that what
we're doing somehow,
00:14:20.330 --> 00:14:24.020
we're taking the determinant
of some matrix that
00:14:24.020 --> 00:14:26.660
has two equal rows.
00:14:26.660 --> 00:14:31.040
So I believe that if we multiply
these by the cofactors from
00:14:31.040 --> 00:14:35.770
some other row, we're taking
the determinant -- ye,
00:14:35.770 --> 00:14:38.660
what matrix are we taking
the determinant of?
00:14:38.660 --> 00:14:40.380
Here it's, this is it.
00:14:40.380 --> 00:14:44.900
We're, when we do that times
this, we're really taking --
00:14:44.900 --> 00:14:48.510
can I put this in little
letters down here?
00:14:48.510 --> 00:14:57.220
I'm taking -- let me look
at the matrix a b a b.
00:14:57.220 --> 00:15:03.610
Let me call that matrix
AS, A screwed up.
00:15:03.610 --> 00:15:04.290
OK.
00:15:04.290 --> 00:15:05.240
All right.
00:15:05.240 --> 00:15:09.260
So now that matrix is
certainly singular.
00:15:09.260 --> 00:15:12.380
So if we find its determinant,
we're going to get zero.
00:15:12.380 --> 00:15:16.810
But I claim that if we find
its determinant by the cofactor
00:15:16.810 --> 00:15:19.960
rule, go along the
first row, we would
00:15:19.960 --> 00:15:23.810
take a times the cofactor of a.
00:15:23.810 --> 00:15:27.180
And what is the --
00:15:27.180 --> 00:15:32.440
see, how -- oh no -- let
me go along the second row.
00:15:32.440 --> 00:15:33.810
OK.
00:15:33.810 --> 00:15:36.320
So let's see, which --
00:15:36.320 --> 00:15:37.190
if I take --
00:15:37.190 --> 00:15:40.510
I know I've got a
singular matrix here.
00:15:40.510 --> 00:15:46.310
And I believe that when I
do this multiplication, what
00:15:46.310 --> 00:15:51.110
I'm doing is using the cofactor
formula for the determinant.
00:15:51.110 --> 00:15:52.700
And I know I'm
going to get zero.
00:15:52.700 --> 00:15:54.530
Let me try this again.
00:15:54.530 --> 00:15:56.920
So the cofactor formula
for the determinant
00:15:56.920 --> 00:16:04.170
says I should take a times
its cofactor, which is this b,
00:16:04.170 --> 00:16:09.260
plus b times its cofactor,
which is this minus a.
00:16:09.260 --> 00:16:10.040
OK.
00:16:10.040 --> 00:16:15.260
That's what we're doing,
apart from a sign here.
00:16:15.260 --> 00:16:20.980
Oh yeah, so you know, there
might be a minus multiplying
00:16:20.980 --> 00:16:21.950
everything.
00:16:21.950 --> 00:16:24.280
So if I take this
determinant, it's A --
00:16:24.280 --> 00:16:29.000
the determinant of this, the
determinant of A screwed up is
00:16:29.000 --> 00:16:33.050
a times its
cofactor, which is b,
00:16:33.050 --> 00:16:38.980
plus the second guy times its
cofactor, which is minus a.
00:16:38.980 --> 00:16:41.790
And of course I get
the answer zero,
00:16:41.790 --> 00:16:45.790
and this is exactly what's
happening in that, in that,
00:16:45.790 --> 00:16:47.650
row times this wrong column.
00:16:47.650 --> 00:16:48.150
OK.
00:16:52.740 --> 00:16:56.460
That's the two by two picture,
and it's just the same here.
00:16:56.460 --> 00:17:01.870
That the reason I get a zero
up in there is, the reason
00:17:01.870 --> 00:17:09.319
I get a zero is that when I
multiply the first row of A
00:17:09.319 --> 00:17:12.880
and the last row of
the cofactor matrix,
00:17:12.880 --> 00:17:16.450
it's as if I'm taking the
determinant of this screwed up
00:17:16.450 --> 00:17:19.420
matrix that has first
and last rows identical.
00:17:22.270 --> 00:17:26.150
The book pins this
down more specific --
00:17:26.150 --> 00:17:29.410
and more carefully than
I can do in the lecture.
00:17:29.410 --> 00:17:31.200
I hope you're seeing the point.
00:17:31.200 --> 00:17:34.130
That this is an identity.
00:17:34.130 --> 00:17:37.310
That it's a beautiful
identity and it tells us what
00:17:37.310 --> 00:17:40.690
the inverse of the matrix is.
00:17:40.690 --> 00:17:43.820
So it gives us the inverse,
the formula for the inverse.
00:17:43.820 --> 00:17:44.510
OK.
00:17:44.510 --> 00:17:49.180
So that's the first goal of
my lecture, was to find this
00:17:49.180 --> 00:17:49.960
formula.
00:17:49.960 --> 00:17:51.490
It's done.
00:17:51.490 --> 00:17:53.800
OK.
00:17:53.800 --> 00:17:58.180
And of course I could
invert, now, I can,
00:17:58.180 --> 00:18:00.900
I sort of like I can see what --
00:18:00.900 --> 00:18:02.850
I can answer
questions like this.
00:18:02.850 --> 00:18:08.510
Suppose I have a matrix,
and let me move the one one
00:18:08.510 --> 00:18:09.740
entry.
00:18:09.740 --> 00:18:13.600
What happens to the inverse?
00:18:13.600 --> 00:18:15.780
Just, just think
about that question.
00:18:15.780 --> 00:18:18.030
Suppose I have some
matrix, I just write down
00:18:18.030 --> 00:18:21.300
some nice, non-singular
matrix that's got an inverse,
00:18:21.300 --> 00:18:25.170
and then I move the one
one entry a little bit.
00:18:25.170 --> 00:18:27.880
I add one to it, for example.
00:18:27.880 --> 00:18:30.730
What happens to
the inverse matrix?
00:18:30.730 --> 00:18:33.160
Well, this formula
should tell me.
00:18:33.160 --> 00:18:36.110
I have to look to see what
happens to the determinant
00:18:36.110 --> 00:18:39.400
and what happens to
all the cofactors.
00:18:39.400 --> 00:18:44.300
And, the picture,
it's all there.
00:18:44.300 --> 00:18:44.960
It's all there.
00:18:44.960 --> 00:18:49.420
We can really understand
how the inverse changes
00:18:49.420 --> 00:18:51.250
when the matrix changes.
00:18:51.250 --> 00:18:52.330
OK.
00:18:52.330 --> 00:18:58.910
Now my second application is to
-- let me put that over here --
00:18:58.910 --> 00:18:59.630
is to Ax=b.
00:19:03.650 --> 00:19:09.060
Well, the -- course, the
solution is A inverse b.
00:19:09.060 --> 00:19:11.820
But now I have a
formula for A inverse.
00:19:11.820 --> 00:19:17.170
A inverse is one
over the determinant
00:19:17.170 --> 00:19:22.030
times this C transpose times B.
00:19:22.030 --> 00:19:24.510
I now know what A inverse is.
00:19:24.510 --> 00:19:27.270
So now I just have to
say, what have I got here?
00:19:27.270 --> 00:19:33.600
Is there any way to, to make
this formula, this answer,
00:19:33.600 --> 00:19:36.550
which is the one
and only answer --
00:19:36.550 --> 00:19:40.980
it's the very same answer we got
on the first day of the class
00:19:40.980 --> 00:19:42.620
by elimination.
00:19:42.620 --> 00:19:47.290
Now I'm -- now I've got
a formula for the answer.
00:19:47.290 --> 00:19:51.890
Can I play with it further
to see what's going on?
00:19:51.890 --> 00:20:01.640
And Cramer's, this Cramer's
Rule is exactly, that --
00:20:01.640 --> 00:20:07.010
a way of looking
at this formula.
00:20:07.010 --> 00:20:07.510
OK.
00:20:07.510 --> 00:20:11.620
So this is a formula for x.
00:20:11.620 --> 00:20:13.620
Here's my formula.
00:20:13.620 --> 00:20:14.350
Well, of course.
00:20:14.350 --> 00:20:16.050
The first thing I
see from the formula
00:20:16.050 --> 00:20:21.100
is that the answer x always
has that in the determinant.
00:20:21.100 --> 00:20:22.590
I'm not surprised.
00:20:22.590 --> 00:20:25.240
There's a division
by the determinant.
00:20:25.240 --> 00:20:29.190
But then I have to say a little
more carefully what's going on
00:20:29.190 --> 00:20:31.610
And let me tell you what
Cramer's Rule is. up here.
00:20:31.610 --> 00:20:35.610
Let, let me take x1,
the first component.
00:20:35.610 --> 00:20:37.740
So this is the first
component of the answer.
00:20:37.740 --> 00:20:40.870
Then there'll be a
second component and a,
00:20:40.870 --> 00:20:42.900
all the other components.
00:20:42.900 --> 00:20:46.310
Can I take just the first
component of this formula?
00:20:46.310 --> 00:20:52.040
Well, I certainly have
determinant of A down under.
00:20:52.040 --> 00:20:56.400
And what the heck
is the first --
00:20:56.400 --> 00:21:00.300
so what do I get
in C transpose b?
00:21:00.300 --> 00:21:03.440
What's the first entry
of C transpose b?
00:21:03.440 --> 00:21:06.060
That's what I have
to answer myself.
00:21:06.060 --> 00:21:10.420
Well, what's the first
entry of C transpose b?
00:21:13.820 --> 00:21:17.670
This B is -- let me
tell you what it is.
00:21:17.670 --> 00:21:18.170
OK.
00:21:18.170 --> 00:21:22.290
Somehow I'm
multiplying cofactors
00:21:22.290 --> 00:21:26.970
by the entries of B,
right, in this product.
00:21:26.970 --> 00:21:30.430
Cofactors from the matrix
times entries of b.
00:21:30.430 --> 00:21:33.650
So any time I'm multiplying
cofactors by numbers,
00:21:33.650 --> 00:21:36.980
I think, I'm getting the
determinant of something.
00:21:36.980 --> 00:21:39.990
And let me call
that something B1.
00:21:39.990 --> 00:21:46.730
So this is a matrix, the matrix
whose determinant is coming out
00:21:46.730 --> 00:21:47.230
of that.
00:21:47.230 --> 00:21:49.530
And we'll, we'll see what it is.
00:21:49.530 --> 00:21:54.550
x2 will be the determinant
of some other matrix B2, also
00:21:54.550 --> 00:21:57.650
divided by determinant of A.
00:21:57.650 --> 00:21:59.190
So now I just --
00:21:59.190 --> 00:22:00.970
Cramer just had a good idea.
00:22:00.970 --> 00:22:06.920
He realized what matrix it was,
what these B1 and B2 and B3
00:22:06.920 --> 00:22:08.470
and so on matrices were.
00:22:08.470 --> 00:22:10.520
Let me write them on
the board underneath.
00:22:14.610 --> 00:22:15.260
OK.
00:22:15.260 --> 00:22:17.470
So what is this B1?
00:22:17.470 --> 00:22:24.430
This B1 is the matrix that
has b in its first column
00:22:24.430 --> 00:22:27.900
and otherwise the
rest of it is A.
00:22:27.900 --> 00:22:38.740
So it otherwise it has the
rest, the, the n-1 columns of A.
00:22:38.740 --> 00:22:42.880
It's the matrix with --
00:22:42.880 --> 00:22:51.410
it's just the matrix A
with column one replaced
00:22:51.410 --> 00:22:58.520
by the right-hand side,
by the right-hand side b.
00:22:58.520 --> 00:23:04.490
Because somehow when I take
the determinant of this guy,
00:23:04.490 --> 00:23:09.140
it's giving me this
matrix multiplication.
00:23:09.140 --> 00:23:10.570
Well, how could that be?
00:23:13.180 --> 00:23:16.490
How could -- so what's,
what's the determinant formula
00:23:16.490 --> 00:23:18.050
I'll use here?
00:23:18.050 --> 00:23:21.910
I'll use cofactors, of course.
00:23:21.910 --> 00:23:25.240
And I might as well use
cofactors down column one.
00:23:25.240 --> 00:23:27.910
So when I use cofactors
down column one,
00:23:27.910 --> 00:23:32.840
I'm taking the first
entry of b times what?
00:23:32.840 --> 00:23:35.650
Times the cofactor c11.
00:23:35.650 --> 00:23:38.410
Do you see that?
00:23:38.410 --> 00:23:40.830
When I, when I use
cofactors here,
00:23:40.830 --> 00:23:43.530
I take the first
entry here, B one
00:23:43.530 --> 00:23:47.120
let's call it, times
the cofactor there.
00:23:47.120 --> 00:23:51.390
But what's the cofactor in --
my little hand-waving is meant
00:23:51.390 --> 00:23:54.650
to indicate that it's a
matrix of one size smaller,
00:23:54.650 --> 00:23:56.030
the cofactor.
00:23:56.030 --> 00:23:58.730
And it's exactly c11.
00:23:58.730 --> 00:24:01.130
Well, that's just
what we wanted.
00:24:01.130 --> 00:24:06.200
This first entry
is c11 times b1.
00:24:06.200 --> 00:24:13.970
And then the next entry is
whatever, is c21 times b2,
00:24:13.970 --> 00:24:14.860
and so on.
00:24:14.860 --> 00:24:17.310
And sure enough, if
I look here, when
00:24:17.310 --> 00:24:19.720
I'm, when I do the
cofactor expansion,
00:24:19.720 --> 00:24:22.790
b2 is getting multiplied by
the right thing, and so on.
00:24:25.450 --> 00:24:27.400
So there's Cramer's Rule.
00:24:27.400 --> 00:24:31.560
And the book gives
another kind of cute proof
00:24:31.560 --> 00:24:36.960
without, without building
so much on, on cofactors.
00:24:36.960 --> 00:24:38.950
But here we've got
cofactors, so I thought
00:24:38.950 --> 00:24:40.650
I'd just give you this proof.
00:24:40.650 --> 00:24:42.700
So what is B --
00:24:42.700 --> 00:24:45.430
in general, what is Bj?
00:24:45.430 --> 00:24:58.970
This is the, this is A with
column j replaced by, by b.
00:25:02.920 --> 00:25:08.870
So that's -- the determinant
of that matrix that you divide
00:25:08.870 --> 00:25:11.700
by determinant of A to get xj.
00:25:11.700 --> 00:25:14.980
So x -- let me change
this general formula.
00:25:14.980 --> 00:25:18.190
xj, the j-th one,
is the determinant
00:25:18.190 --> 00:25:22.510
of Bj divided by the
determinant of A.
00:25:22.510 --> 00:25:24.360
And now we've said what Bj is.
00:25:30.160 --> 00:25:33.720
Well, so Cramer found a rule.
00:25:33.720 --> 00:25:39.320
And we could ask him, OK,
great, good work, Cramer.
00:25:39.320 --> 00:25:44.180
But is your rule any
good in practice?
00:25:44.180 --> 00:25:49.960
So he says, well, you couldn't
ask about a rule in mine,
00:25:49.960 --> 00:25:52.220
right, because it's just --
00:25:52.220 --> 00:25:56.850
all you have to do is find
the determinant of A and these
00:25:56.850 --> 00:26:01.220
other determinants, so I guess
-- oh, he just says, well,
00:26:01.220 --> 00:26:04.140
all you have to do is
find n+1 determinants,
00:26:04.140 --> 00:26:06.170
the, the n Bs and the A.
00:26:06.170 --> 00:26:17.530
And actually, I remember
reading -- there was a book,
00:26:17.530 --> 00:26:22.530
popular book that, that kids
interested in math read when I
00:26:22.530 --> 00:26:25.980
was a kid interested in
math called Mathematics
00:26:25.980 --> 00:26:29.840
for the Million or something,
by a guy named Bell.
00:26:29.840 --> 00:26:34.830
And it had a little page
about linear algebra.
00:26:34.830 --> 00:26:39.000
And it said,-- so it
explained elimination
00:26:39.000 --> 00:26:41.250
in a very complicated way.
00:26:41.250 --> 00:26:43.200
I certainly didn't
understand it.
00:26:43.200 --> 00:26:47.950
And, and it made it, you
know, it sort of said, well,
00:26:47.950 --> 00:26:51.120
there is this formula
for elimination,
00:26:51.120 --> 00:26:55.770
but look at this great
formula, Cramer's Rule.
00:26:55.770 --> 00:27:00.410
So it certainly said Cramer's
Rule was the way to go.
00:27:00.410 --> 00:27:05.180
But actually, Cramer's Rule
is a disastrous way to go,
00:27:05.180 --> 00:27:07.920
because to compute
these determinants,
00:27:07.920 --> 00:27:12.060
it takes, like,
approximately forever.
00:27:12.060 --> 00:27:17.020
So actually I now think
of that book title
00:27:17.020 --> 00:27:18.980
as being Mathematics
for the Millionaire,
00:27:18.980 --> 00:27:22.160
because you'd have to
be able to pay for,
00:27:22.160 --> 00:27:26.570
a hopelessly long calculation
where elimination, of course,
00:27:26.570 --> 00:27:30.450
produced the x-s, in an instant.
00:27:30.450 --> 00:27:36.060
But having a formula allows
you to, with, with letters, you
00:27:36.060 --> 00:27:40.010
know, allows you to do algebra
instead of, algorithms.
00:27:40.010 --> 00:27:44.220
So the, there's some value
in the Cramer's Rule formula
00:27:44.220 --> 00:27:52.000
for x and in the explicit
formula for, for A inverse.
00:27:52.000 --> 00:27:54.890
They're nice
formulas, but I just
00:27:54.890 --> 00:27:57.710
don't want you to use them.
00:27:57.710 --> 00:27:59.230
That'ss what it comes to.
00:27:59.230 --> 00:28:02.960
If you had to -- and Matlab
would never, never do it.
00:28:02.960 --> 00:28:05.480
I mean, it would
use elimination.
00:28:05.480 --> 00:28:06.720
OK.
00:28:06.720 --> 00:28:11.380
Now I'm ready for number
three in today's list
00:28:11.380 --> 00:28:17.790
of amazing connections coming
through the determinant.
00:28:17.790 --> 00:28:21.900
And that number three is the
fact that the determinant gives
00:28:21.900 --> 00:28:22.950
a volume.
00:28:22.950 --> 00:28:24.480
OK.
00:28:24.480 --> 00:28:28.860
So now -- so that's
my final topic for --
00:28:28.860 --> 00:28:33.070
among these -- this my
number three application,
00:28:33.070 --> 00:28:37.700
that the determinant is actually
equals the volume of something.
00:28:37.700 --> 00:28:42.720
Can I use this little space
to consider a special case,
00:28:42.720 --> 00:28:46.050
and then I'll use the
far board to think
00:28:46.050 --> 00:28:47.840
about the general rule.
00:28:47.840 --> 00:28:50.400
So what I going to prove?
00:28:50.400 --> 00:28:51.970
Or claim.
00:28:51.970 --> 00:28:56.950
I claim that the
determinant of the matrix
00:28:56.950 --> 00:29:02.800
is the volume of a box.
00:29:02.800 --> 00:29:05.930
OK, and you say, which box?
00:29:05.930 --> 00:29:06.970
Fair enough.
00:29:06.970 --> 00:29:08.570
OK.
00:29:08.570 --> 00:29:11.670
So let's see.
00:29:11.670 --> 00:29:16.170
I'm in -- shall we say we're
in, say three by three?
00:29:16.170 --> 00:29:18.780
Shall we suppose -- let's,
let's say three by three.
00:29:18.780 --> 00:29:22.680
So, so we can really -- we're,
we're talking about boxes
00:29:22.680 --> 00:29:26.400
in three dimensions, and
three by three matrices.
00:29:26.400 --> 00:29:30.990
And so all I do -- you
could guess what the box is.
00:29:30.990 --> 00:29:34.700
Here is, here is,
three dimensions.
00:29:34.700 --> 00:29:35.450
OK.
00:29:35.450 --> 00:29:41.880
Now I take the first row of
the matrix, a11, a22, A --
00:29:41.880 --> 00:29:44.360
sorry. a11, a12, a13.
00:29:44.360 --> 00:29:48.090
That row is a vector.
00:29:48.090 --> 00:29:50.190
It goes to some point.
00:29:50.190 --> 00:29:53.220
That point will be -- and
that edge going to it,
00:29:53.220 --> 00:29:56.650
will be an edge of the box,
and that point will be a corner
00:29:56.650 --> 00:29:57.740
of the box.
00:29:57.740 --> 00:30:01.010
So here is zero zero
zero, of course.
00:30:01.010 --> 00:30:09.500
And here's the first row of
the matrix: a11, a12, a13.
00:30:09.500 --> 00:30:14.840
So that's one edge of the box.
00:30:14.840 --> 00:30:19.700
Another edge of the box
is to the second row
00:30:19.700 --> 00:30:21.820
of the matrix, row two.
00:30:21.820 --> 00:30:24.510
Can I just call
it there row two?
00:30:24.510 --> 00:30:28.760
And a third row of
the box will be to --
00:30:28.760 --> 00:30:32.200
a third row -- a third edge of
the box will be given by row
00:30:32.200 --> 00:30:33.300
three.
00:30:33.300 --> 00:30:36.470
So, so there's row three.
00:30:36.470 --> 00:30:38.780
That, the coordinates,
what are the coordinates
00:30:38.780 --> 00:30:41.030
of that corner of the box?
00:30:41.030 --> 00:30:47.650
a31, a32, a33.
00:30:47.650 --> 00:30:51.220
So I've got that edge of the
box, that edge of the box,
00:30:51.220 --> 00:30:53.830
that edge of the box,
and that's all I need.
00:30:53.830 --> 00:30:59.820
Now I just finish
out the box, right?
00:30:59.820 --> 00:31:03.650
I just -- the proper word,
of course, is parallelepiped.
00:31:03.650 --> 00:31:08.920
But for obvious
reasons, I wrote box.
00:31:08.920 --> 00:31:09.550
OK.
00:31:09.550 --> 00:31:11.360
So, OK.
00:31:11.360 --> 00:31:14.550
So there's the, there's
the bottom of the box.
00:31:14.550 --> 00:31:19.660
There're the four
edge sides of the box.
00:31:19.660 --> 00:31:23.570
There's the top of the box.
00:31:23.570 --> 00:31:24.540
Cute, right?
00:31:24.540 --> 00:31:28.470
It's the box that
has these three edges
00:31:28.470 --> 00:31:33.080
and then it's completed to
a, to a, each, you know,
00:31:33.080 --> 00:31:35.080
each side is a, is
a parallelogram.
00:31:37.730 --> 00:31:43.050
And it's that box whose volume
is given by the determinant.
00:31:45.610 --> 00:31:50.650
That's -- now it's -- possible
that the determinant is
00:31:50.650 --> 00:31:52.920
negative.
00:31:52.920 --> 00:31:56.110
So we have to just say
what's going on in that case.
00:31:56.110 --> 00:32:01.840
If the determinant is
negative, then the volume, we,
00:32:01.840 --> 00:32:04.030
we should take the
absolute value really.
00:32:04.030 --> 00:32:07.030
So the volume, if we, if we
think of volume as positive,
00:32:07.030 --> 00:32:11.310
we should take the absolute
value of the determinant.
00:32:11.310 --> 00:32:14.400
But the, the sign, what does
the sign of the determinant --
00:32:14.400 --> 00:32:16.560
it always must
tell us something.
00:32:16.560 --> 00:32:20.380
And somehow it, it will tell
us whether these three is a --
00:32:20.380 --> 00:32:23.270
whether it's a right-handed
box or a left-handed box.
00:32:23.270 --> 00:32:30.300
If we, if we reversed
two of the edges,
00:32:30.300 --> 00:32:32.250
we would go between
a right-handed box
00:32:32.250 --> 00:32:33.390
and a left-handed box.
00:32:33.390 --> 00:32:35.440
We wouldn't change the
volume, but we would
00:32:35.440 --> 00:32:40.500
change the, the cyclic, order.
00:32:40.500 --> 00:32:42.900
So I won't worry about that.
00:32:42.900 --> 00:32:47.020
And, so one special
case is what?
00:32:47.020 --> 00:32:49.930
A equal identity matrix.
00:32:49.930 --> 00:32:52.070
So let's take that special case.
00:32:52.070 --> 00:32:55.520
A equal identity matrix.
00:32:55.520 --> 00:32:59.550
Is the formula determinant
of identity matrix,
00:32:59.550 --> 00:33:01.230
does that equal the
volume of the box?
00:33:04.300 --> 00:33:06.460
Well, what is the box?
00:33:06.460 --> 00:33:07.370
What's the box?
00:33:07.370 --> 00:33:12.230
If A is the identity matrix,
then these three rows are
00:33:12.230 --> 00:33:17.460
the three coordinate
vectors, and the box is --
00:33:17.460 --> 00:33:18.920
it's a cube.
00:33:18.920 --> 00:33:21.200
It's the unit cube.
00:33:21.200 --> 00:33:23.970
So if, if A is the
identity matrix, of course
00:33:23.970 --> 00:33:26.080
our formula is
00:33:26.080 --> 00:33:29.850
Well, actually that
proves property one --
00:33:29.850 --> 00:33:31.740
that the volume right.
has property one.
00:33:31.740 --> 00:33:35.440
Actually, we could, we could,
we could get this thing if we --
00:33:35.440 --> 00:33:39.210
if we can show that the box
volume has the same three
00:33:39.210 --> 00:33:42.010
properties that define
the determinant,
00:33:42.010 --> 00:33:44.060
then it must be the determinant.
00:33:46.670 --> 00:33:51.080
So that's like the, the, the
elegant way to prove this.
00:33:51.080 --> 00:33:53.860
To prove this amazing fact
that the determinant equals
00:33:53.860 --> 00:33:58.540
the volume, first we'll check
it for the identity matrix.
00:33:58.540 --> 00:34:00.470
That's fine.
00:34:00.470 --> 00:34:03.190
The box is a cube
and its volume is one
00:34:03.190 --> 00:34:07.180
and the determinant is one
and, and one agrees with one.
00:34:07.180 --> 00:34:11.590
Now let me take one -- let me
go up one level to an orthogonal
00:34:11.590 --> 00:34:12.889
matrix.
00:34:12.889 --> 00:34:16.060
Because I'd like to take this
chance to bring in chapter --
00:34:16.060 --> 00:34:18.750
the, the previous chapter.
00:34:18.750 --> 00:34:21.100
Suppose I have an
orthogonal matrix.
00:34:21.100 --> 00:34:22.110
What did that mean?
00:34:22.110 --> 00:34:25.500
I always called those things Q.
00:34:25.500 --> 00:34:27.909
What was the point
of -- suppose I have,
00:34:27.909 --> 00:34:32.670
suppose instead of the identity
matrix I'm now going to take A
00:34:32.670 --> 00:34:36.910
equal Q, an orthogonal matrix.
00:34:42.940 --> 00:34:46.510
What was Q then?
00:34:46.510 --> 00:34:52.179
That was a matrix whose columns
were orthonormal, right?
00:34:52.179 --> 00:34:54.610
Those were its columns
were unit vectors,
00:34:54.610 --> 00:34:56.059
perpendicular unit vectors.
00:34:59.220 --> 00:35:03.090
So what kind of a
box have we got now?
00:35:03.090 --> 00:35:06.520
What kind of a box comes
from the rows or the columns,
00:35:06.520 --> 00:35:08.530
I don't mind, because
the determinant
00:35:08.530 --> 00:35:10.420
is the determinant
of the transpose,
00:35:10.420 --> 00:35:11.850
so I'm never worried about
00:35:11.850 --> 00:35:12.730
that.
00:35:12.730 --> 00:35:15.850
What kind of a box,
what shape box have we
00:35:15.850 --> 00:35:20.380
got if the matrix is
an orthogonal matrix?
00:35:20.380 --> 00:35:23.040
It's another cube.
00:35:23.040 --> 00:35:24.710
It's a cube again.
00:35:24.710 --> 00:35:27.900
How is it different
from the identity cube?
00:35:30.510 --> 00:35:33.140
It's just rotated.
00:35:33.140 --> 00:35:36.830
It's just the orthogonal
matrix Q doesn't
00:35:36.830 --> 00:35:38.860
have to be the identity matrix.
00:35:38.860 --> 00:35:43.240
It's just the unit cube
but turned in space.
00:35:43.240 --> 00:35:48.490
So sure enough, it's the unit
cube, and its volume is one.
00:35:48.490 --> 00:35:53.410
Now is the determinant one?
00:35:53.410 --> 00:35:55.410
What's the determinant of Q?
00:35:55.410 --> 00:35:58.980
We believe that the determinant
of Q better be one or minus
00:35:58.980 --> 00:36:03.100
one, so that our formula
is -- checks out in that --
00:36:03.100 --> 00:36:06.800
if we can't check it in these
easy cases where we got a cube,
00:36:06.800 --> 00:36:11.020
we're not going to get
it in the general case.
00:36:11.020 --> 00:36:18.290
So why is the determinant
of Q equal one or minus one?
00:36:18.290 --> 00:36:19.810
What do we know about Q?
00:36:19.810 --> 00:36:25.050
What's the one matrix statement
of the properties of Q?
00:36:25.050 --> 00:36:29.020
A matrix with orthonormal
columns has --
00:36:29.020 --> 00:36:31.860
satisfies a certain equation.
00:36:31.860 --> 00:36:33.400
What, what is that?
00:36:33.400 --> 00:36:38.930
It's if we have this orthogonal
matrix, then the fact --
00:36:38.930 --> 00:36:45.550
the way to say what it, what
its properties are is this.
00:36:45.550 --> 00:36:50.940
Q prime, u- u- Q
transpose Q equals I.
00:36:50.940 --> 00:36:52.790
Right?
00:36:52.790 --> 00:36:57.450
That's what -- those are the
matrices that get the name Q,
00:36:57.450 --> 00:37:01.241
the matrices that
Q transpose Q is I.
00:37:01.241 --> 00:37:01.740
OK.
00:37:01.740 --> 00:37:07.180
Now from that, tell me
why is the determinant one
00:37:07.180 --> 00:37:09.500
or minus one.
00:37:09.500 --> 00:37:12.930
How do I, out of this fact --
00:37:12.930 --> 00:37:14.525
this may even be a
homework problem.
00:37:17.240 --> 00:37:20.890
It's there in the, in the
list of exercises in the book,
00:37:20.890 --> 00:37:23.100
and let's just do it.
00:37:23.100 --> 00:37:27.340
How do I get, how do I discover
that the determinant of Q
00:37:27.340 --> 00:37:31.530
is one or maybe minus one?
00:37:31.530 --> 00:37:34.180
I take determinants of
both sides, everybody says,
00:37:34.180 --> 00:37:36.870
so I won't --
00:37:36.870 --> 00:37:38.610
I take determinants
of both sides.
00:37:38.610 --> 00:37:41.250
On the right-hand side -- so
I, when I take determinants
00:37:41.250 --> 00:37:46.240
of both sides,
let me just do it.
00:37:46.240 --> 00:37:48.315
Take the determinant of
-- take determinants.
00:37:51.680 --> 00:37:53.970
Determinant of the
identity is one.
00:37:53.970 --> 00:37:55.840
What's the determinant
of that product?
00:37:58.690 --> 00:38:02.400
Rule nine is paying off now.
00:38:02.400 --> 00:38:07.920
The determinant of a product is
the determinant of this guy --
00:38:07.920 --> 00:38:11.110
maybe I'll put it, I'll use
that symbol for determinant.
00:38:11.110 --> 00:38:14.152
It's the determinant of that
guy times the determinant
00:38:14.152 --> 00:38:14.860
of the other guy.
00:38:17.460 --> 00:38:21.400
And then what's the
determinant of Q transpose?
00:38:21.400 --> 00:38:22.990
It's the same as the
determinant of Q.
00:38:22.990 --> 00:38:24.810
Rule ten pays off.
00:38:24.810 --> 00:38:28.330
So this is just
this thing squared.
00:38:28.330 --> 00:38:33.740
So that determinant squared is
one and sure enough it's one
00:38:33.740 --> 00:38:35.290
or minus one.
00:38:35.290 --> 00:38:35.790
Great.
00:38:38.370 --> 00:38:42.000
So in these special
cases of cubes,
00:38:42.000 --> 00:38:48.650
we really do have
determinant equals volume.
00:38:48.650 --> 00:38:54.530
Now can I just push
that to non-cubes.
00:38:54.530 --> 00:39:02.930
Let me push it first to
rectangles, rectangular boxes,
00:39:02.930 --> 00:39:07.820
where I'm just multiplying
the e- the edges are --
00:39:07.820 --> 00:39:10.440
let me keep all the
ninety degree angles,
00:39:10.440 --> 00:39:13.510
because those are -- that,
that makes my life easy.
00:39:13.510 --> 00:39:16.910
And just stretch the edges.
00:39:16.910 --> 00:39:21.910
Suppose I stretch that first
edge, suppose this first edge
00:39:21.910 --> 00:39:23.230
I double.
00:39:23.230 --> 00:39:27.990
Suppose I double
that first edge,
00:39:27.990 --> 00:39:31.060
keeping the other
edges the same.
00:39:31.060 --> 00:39:34.650
What happens to the volume?
00:39:34.650 --> 00:39:36.640
It doubles, right?
00:39:36.640 --> 00:39:39.420
We know that the volume
of a cube doubles.
00:39:39.420 --> 00:39:42.370
In fact, because we know that
the new cube would sit right
00:39:42.370 --> 00:39:43.400
on top --
00:39:43.400 --> 00:39:46.210
I mean, the new, the added
cube would sit right on --
00:39:46.210 --> 00:39:47.620
would fit --
00:39:47.620 --> 00:39:50.120
probably a geometer would
say congruent or something --
00:39:50.120 --> 00:39:51.810
would go right in, in the other.
00:39:51.810 --> 00:39:52.830
We'd have two.
00:39:52.830 --> 00:39:54.540
We have two identical cubes.
00:39:54.540 --> 00:39:58.421
Total volume is now two.
00:39:58.421 --> 00:39:58.920
OK.
00:39:58.920 --> 00:40:02.392
So I want -- if I double an
edge, the volume doubles.
00:40:02.392 --> 00:40:03.725
What happens to the determinant?
00:40:07.580 --> 00:40:14.880
If I double, the first
row of a matrix, what ch-
00:40:14.880 --> 00:40:18.010
ch- what's the effect
on the determinant?
00:40:18.010 --> 00:40:21.320
It also doubles, right?
00:40:21.320 --> 00:40:25.910
And that was rule number 3a.
00:40:25.910 --> 00:40:29.500
Remember rule 3a was
that if I, I could,
00:40:29.500 --> 00:40:36.780
if I had a factor in, in row
one, T, I could factor it out.
00:40:36.780 --> 00:40:41.020
So if, if I have a factor
two in that row one,
00:40:41.020 --> 00:40:43.340
I can factor it out
of the determinant.
00:40:43.340 --> 00:40:48.860
It agrees with the -- the volume
of the box has that factor two.
00:40:48.860 --> 00:40:53.140
So, so volume satisfies
this property 3a.
00:40:53.140 --> 00:40:59.040
And now I really close, but I
-- but to get to the very end
00:40:59.040 --> 00:41:01.980
of this proof, I have
to get away from right
00:41:01.980 --> 00:41:02.650
angles.
00:41:02.650 --> 00:41:10.330
I have to allow the
possibility of, other angles.
00:41:10.330 --> 00:41:13.090
And -- or what's
saying the same thing,
00:41:13.090 --> 00:41:18.600
I have to check that the
volume also satisfies 3b.
00:41:18.600 --> 00:41:20.600
So can I --
00:41:20.600 --> 00:41:24.740
This is end of proof
that the -- so I'm --
00:41:24.740 --> 00:41:34.380
determinant of A equals volume
of box, and where I right now?
00:41:34.380 --> 00:41:40.820
This volume has properties,
properties one, no problem.
00:41:40.820 --> 00:41:44.030
If the box is the
cube, everything is --
00:41:44.030 --> 00:41:49.000
if the box is the unit
cube, its volume is one.
00:41:49.000 --> 00:41:54.680
Property two was if
I reverse two rows,
00:41:54.680 --> 00:41:57.960
but that doesn't change the box.
00:41:57.960 --> 00:42:00.520
And it doesn't change the
absolute value, so no problem
00:42:00.520 --> 00:42:01.200
there.
00:42:01.200 --> 00:42:06.890
Property 3a was if I mul-
you remember what 3a was?
00:42:06.890 --> 00:42:10.390
So property one was about
the identity matrix.
00:42:10.390 --> 00:42:12.900
Property two was about
a plus or minus sign
00:42:12.900 --> 00:42:14.560
that I don't care about.
00:42:14.560 --> 00:42:18.490
Property 3a was a
factor T in a row.
00:42:18.490 --> 00:42:24.080
But now I've got property
three B to deal with.
00:42:24.080 --> 00:42:25.520
What was property 3b?
00:42:25.520 --> 00:42:30.270
This is a great way to
review these, properties.
00:42:30.270 --> 00:42:34.860
So that 3b, the property
3b said -- let's do,
00:42:34.860 --> 00:42:36.910
let's do two by two.
00:42:36.910 --> 00:42:42.620
So said that if I
had a+a', b+b', c,
00:42:42.620 --> 00:42:47.590
d that this equaled what?
00:42:47.590 --> 00:42:49.120
So this is property 3b.
00:42:49.120 --> 00:42:53.840
This is the linearity
in row one by itself.
00:42:53.840 --> 00:42:59.060
So c d is staying the same,
and I can split this into a b
00:42:59.060 --> 00:43:01.340
and a' b'.
00:43:06.160 --> 00:43:12.890
That's property 3b, at least
in the two by two case.
00:43:12.890 --> 00:43:15.340
And what I --
00:43:15.340 --> 00:43:20.540
I wanted now to show
that the volume, which
00:43:20.540 --> 00:43:25.040
two, two by two, that
means area, has this,
00:43:25.040 --> 00:43:25.790
has this property.
00:43:28.920 --> 00:43:32.540
Let me just emphasize that we
have got -- we're getting --
00:43:32.540 --> 00:43:37.150
this is a formula, then, for
the area of a parallelogram.
00:43:37.150 --> 00:43:40.310
The area of this parallelogram
-- can I just draw it?
00:43:40.310 --> 00:43:42.160
OK, here's the, here's
the parallelogram.
00:43:42.160 --> 00:43:45.370
I have the row a b.
00:43:45.370 --> 00:43:47.000
That's the first row.
00:43:47.000 --> 00:43:49.260
That's the point a b.
00:43:49.260 --> 00:43:54.060
And I tack on c d.
00:43:54.060 --> 00:43:57.690
c d, coming out of here.
00:43:57.690 --> 00:43:59.065
And I complete
the parallelogram.
00:44:02.560 --> 00:44:03.750
So this is --
00:44:03.750 --> 00:44:08.140
well, I better
make it look right.
00:44:08.140 --> 00:44:12.530
It's really this one that has
coordinates c d and this has
00:44:12.530 --> 00:44:17.040
coordinates -- well,
whatever the sum is.
00:44:17.040 --> 00:44:18.820
And of course
starting at zero zero.
00:44:22.240 --> 00:44:26.026
So we all know,
this is a+c, b+d.
00:44:29.140 --> 00:44:31.120
Rather than --
00:44:31.120 --> 00:44:33.460
I'm pausing on that
proof for a minute
00:44:33.460 --> 00:44:37.320
just to going back
to our formula.
00:44:37.320 --> 00:44:40.650
Because I want you to see
that unlike Cramer's Rule,
00:44:40.650 --> 00:44:44.060
that I wasn't that
impressed by, I'm
00:44:44.060 --> 00:44:46.640
very impressed by this
formula for the area
00:44:46.640 --> 00:44:48.540
of a parallelogram.
00:44:48.540 --> 00:44:50.930
And what's our formula?
00:44:50.930 --> 00:44:54.890
What, what's the area
of that parallelogram?
00:44:54.890 --> 00:45:00.250
If I had asked you
that last year,
00:45:00.250 --> 00:45:03.950
you would have said OK,
the area of a parallelogram
00:45:03.950 --> 00:45:06.060
is the base times the height,
00:45:06.060 --> 00:45:07.190
right?
00:45:07.190 --> 00:45:10.890
So you would have figured
out what this base, the --
00:45:10.890 --> 00:45:12.850
how long that base was.
00:45:12.850 --> 00:45:16.182
It's like the square root
of A squared plus b squared.
00:45:16.182 --> 00:45:17.640
And then you would
have figured out
00:45:17.640 --> 00:45:20.360
how much is this
height, whatever it is.
00:45:20.360 --> 00:45:21.920
It's horrible.
00:45:21.920 --> 00:45:28.330
This, I mean, we got square
roots, and in that height
00:45:28.330 --> 00:45:32.040
there would be other
revolting stuff.
00:45:32.040 --> 00:45:35.800
But now what's the formula
that we now know for the area?
00:45:42.290 --> 00:45:45.740
It's the determinant
of our little matrix.
00:45:48.540 --> 00:45:51.050
It's just ad-bc.
00:45:56.990 --> 00:45:59.110
No square roots.
00:45:59.110 --> 00:46:03.000
Totally rememberable, because
it's exactly a formula
00:46:03.000 --> 00:46:06.960
that we've been studying the
whole, for three lectures.
00:46:06.960 --> 00:46:07.460
OK.
00:46:11.090 --> 00:46:13.960
That's, you know, that's
the most important point
00:46:13.960 --> 00:46:15.190
I'm making here.
00:46:15.190 --> 00:46:20.990
Is that if you know the
coordinates of a box,
00:46:20.990 --> 00:46:24.750
of the corners, then
you have a great formula
00:46:24.750 --> 00:46:28.620
for the volume,
area or volume, that
00:46:28.620 --> 00:46:33.550
doesn't involve any lengths
or any angles or any heights,
00:46:33.550 --> 00:46:37.850
but just involves the
coordinates that you've got.
00:46:37.850 --> 00:46:40.390
And similarly, what's the
area of this triangle?
00:46:40.390 --> 00:46:43.530
Suppose I chop that off
and say what about --
00:46:43.530 --> 00:46:45.810
because you might often be
interested in a triangle
00:46:45.810 --> 00:46:47.330
instead of a parallelogram.
00:46:47.330 --> 00:46:48.725
What's the area
of this triangle?
00:46:52.040 --> 00:46:53.950
Now there again,
everybody would have
00:46:53.950 --> 00:46:58.110
said the area of a triangle is
half the base times the height.
00:47:01.200 --> 00:47:04.300
And in some cases, if you know
the base that a, that's --
00:47:04.300 --> 00:47:06.300
and the height, that's fine.
00:47:06.300 --> 00:47:10.280
But here, we, what we know is
the coordinates of the corners.
00:47:10.280 --> 00:47:11.980
We know the vertices.
00:47:11.980 --> 00:47:14.690
And so what's the
area of that triangle?
00:47:17.830 --> 00:47:22.550
If I know these, if I know
a b, c d, and zero zero,
00:47:22.550 --> 00:47:25.060
what's the area?
00:47:25.060 --> 00:47:29.670
It's just half, so
it's just half of this.
00:47:29.670 --> 00:47:33.040
So this is, this
is a- a b -- a d -
00:47:33.040 --> 00:47:38.870
b c for the parallelogram
and one half of that,
00:47:38.870 --> 00:47:43.585
one half of ad-bc
for the triangle.
00:47:47.140 --> 00:47:50.900
So I mean, this is a totally
trivial remark, to say, well,
00:47:50.900 --> 00:47:52.630
divide by two.
00:47:52.630 --> 00:47:56.910
But it's just that you
more often see triangles,
00:47:56.910 --> 00:48:01.530
and you feel you know
the formula for the area
00:48:01.530 --> 00:48:04.950
but the good formula for
the area is this one.
00:48:04.950 --> 00:48:06.760
And I'm just going to --
00:48:06.760 --> 00:48:08.260
I'm just going to
say one more thing
00:48:08.260 --> 00:48:09.750
about the area of a triangle.
00:48:09.750 --> 00:48:11.220
It's just because
it's -- you know,
00:48:11.220 --> 00:48:15.760
it's so great to have a
good formula for something.
00:48:15.760 --> 00:48:20.990
What if our triangle did
not start at zero zero?
00:48:20.990 --> 00:48:25.630
What if our triangle,
what if we had this --
00:48:25.630 --> 00:48:28.390
what if we had -- so I'm
coming back to triangles again.
00:48:32.600 --> 00:48:40.320
But let me, let me put this
triangle somewhere, it's --
00:48:40.320 --> 00:48:43.800
I'm staying with triangles,
I'm just in two dimensions,
00:48:43.800 --> 00:48:53.280
but I'm going to allow you
to give me any three corners.
00:48:57.420 --> 00:49:01.430
And in -- those six numbers
must determine the area.
00:49:01.430 --> 00:49:03.750
And what's the formula?
00:49:03.750 --> 00:49:05.280
The area is going
to be, it's going
00:49:05.280 --> 00:49:09.530
to be, there'll be that
half of a parallelogram.
00:49:09.530 --> 00:49:13.580
I mean, basically this can't
be completely new, right?
00:49:13.580 --> 00:49:17.230
We've got the area when -- we,
we know the area when this is
00:49:17.230 --> 00:49:20.400
zero zero.
00:49:20.400 --> 00:49:24.810
Now we just want to lift our
sight slightly and get the area
00:49:24.810 --> 00:49:27.820
when all th- so let me
write down what it, what it
00:49:27.820 --> 00:49:29.190
comes out to be.
00:49:29.190 --> 00:49:37.390
It turns out that if you do
this, x1 y1 and a 1, x2 y2
00:49:37.390 --> 00:49:44.150
and a 1, x3 y3 and a
1, that that works.
00:49:44.150 --> 00:49:47.320
That the determinant
symbol, of course.
00:49:47.320 --> 00:49:51.640
It's just -- if I gave you
that determinant to find,
00:49:51.640 --> 00:49:53.890
you might subtract
this row from this.
00:49:53.890 --> 00:49:55.780
It would kill that one.
00:49:55.780 --> 00:49:59.170
Subtract this row from this,
it would kill that one.
00:49:59.170 --> 00:50:02.470
Then you'd have a simple
determinant to do with
00:50:02.470 --> 00:50:06.300
differences, and it would --
00:50:06.300 --> 00:50:08.960
this little
subtraction, what I did
00:50:08.960 --> 00:50:12.580
was equivalent to
moving the triangle
00:50:12.580 --> 00:50:16.550
to start at the origin.
00:50:16.550 --> 00:50:19.850
I did it fast,
because time is up.
00:50:19.850 --> 00:50:24.530
And I didn't complete
that proof of 3b.
00:50:24.530 --> 00:50:28.930
I'll leave -- the book has a
carefully drawn figure to show
00:50:28.930 --> 00:50:30.870
why that works.
00:50:30.870 --> 00:50:34.110
But I hope you saw
the main point is
00:50:34.110 --> 00:50:36.760
that for area and
volume, determinant
00:50:36.760 --> 00:50:39.540
gives a great formula.
00:50:39.540 --> 00:50:40.100
OK.
00:50:40.100 --> 00:50:44.410
And next lectures are
about eigenvalues,
00:50:44.410 --> 00:50:47.780
so we're really
into the big stuff.
00:50:47.780 --> 00:50:49.330
Thanks.