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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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00:00:35,250 --> 00:00:40,710
packs all this information
into a single number.
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And that number can
give us formulas
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00:00:45,800 --> 00:00:50,510
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?
45
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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?
48
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Because this c was in a two one
position, and two plus one is
49
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odd.
50
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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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00:10:25,700 --> 00:10:32,900
That's the cofactor formula
for the determinant.
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00:10:32,900 --> 00:10:35,240
That's the, this
cofactor formula,
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which I wrote, which
we got last time.
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00:10:39,110 --> 00:10:42,700
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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00:10:51,600 --> 00:10:54,650
And that gives me
the determinant.
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00:10:54,650 --> 00:10:59,470
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.
184
00:11:08,800 --> 00:11:12,830
So up here in this,
in this position,
185
00:11:12,830 --> 00:11:15,470
I'm getting determinant of A.
186
00:11:15,470 --> 00:11:18,130
What about in the
two two position?
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00:11:18,130 --> 00:11:23,460
Row two times column two
there, what, what is that?
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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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00:11:28,960 --> 00:11:32,980
So of course I get
the determinant again.
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00:11:32,980 --> 00:11:36,310
And in the last here,
this is the last row
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00:11:36,310 --> 00:11:37,780
times its cofactors.
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00:11:37,780 --> 00:11:41,540
It's exactly -- you see, we're
realizing that the cofactor
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00:11:41,540 --> 00:11:44,780
formula is just this
sum of products,
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00:11:44,780 --> 00:11:47,160
so of course we think,
hey, we've got a matrix
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00:11:47,160 --> 00:11:48,510
multiplication there.
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00:11:48,510 --> 00:11:57,870
And we get determinant of A.
198
00:11:57,870 --> 00:12:00,920
But there's one more
idea here, right?
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00:12:00,920 --> 00:12:01,990
Great.
200
00:12:01,990 --> 00:12:05,410
What else, what have I not --
so I haven't got that formula
201
00:12:05,410 --> 00:12:10,520
completely proved yet, because
I've still got to do all
202
00:12:10,520 --> 00:12:14,130
the off-diagonal stuff,
which I want to be zero,
203
00:12:14,130 --> 00:12:15,120
right?
204
00:12:15,120 --> 00:12:17,560
I just want this
to be determinant
205
00:12:17,560 --> 00:12:22,850
of A times the identity, and
then I'm, I'm a happy person.
206
00:12:22,850 --> 00:12:24,860
So why should that be?
207
00:12:24,860 --> 00:12:30,660
Why should it be that one
row times the cofactors
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00:12:30,660 --> 00:12:34,920
from a different row, which
become a column because I
209
00:12:34,920 --> 00:12:39,360
transpose, give zero?
210
00:12:39,360 --> 00:12:44,210
In other words, the cofactor
formula gives the determinant
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00:12:44,210 --> 00:12:48,440
if the row and the, and
the cofactors -- you know,
212
00:12:48,440 --> 00:12:54,050
if the entries of A and the
cofactors are for the same row.
213
00:12:54,050 --> 00:12:58,470
But for some reason, if I
take the cofactors from the --
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00:12:58,470 --> 00:13:01,920
entries from the first row and
the cofactors from the second
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00:13:01,920 --> 00:13:04,290
row, for some reason
I automatically
216
00:13:04,290 --> 00:13:05,270
get zero.
217
00:13:05,270 --> 00:13:08,140
And it's sort of
like interesting
218
00:13:08,140 --> 00:13:10,540
to say, why does that happen?
219
00:13:10,540 --> 00:13:14,620
And can I just check that --
of course, we know it happens,
220
00:13:14,620 --> 00:13:16,410
in this case.
221
00:13:16,410 --> 00:13:19,060
Here are the
numbers from row one
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00:13:19,060 --> 00:13:24,600
and here are the cofactors
from row two, right?
223
00:13:24,600 --> 00:13:26,820
Those are the
numbers in row one.
224
00:13:26,820 --> 00:13:29,510
And th- these are the
cofactors from row two,
225
00:13:29,510 --> 00:13:32,110
because the cofactor
of c is minus b
226
00:13:32,110 --> 00:13:33,530
and the cofactor of d is
227
00:13:33,530 --> 00:13:34,030
a.
228
00:13:34,030 --> 00:13:40,380
And sure enough, that row
times this column gives --
229
00:13:40,380 --> 00:13:41,840
please say it.
230
00:13:41,840 --> 00:13:43,370
Zero, right.
231
00:13:43,370 --> 00:13:43,870
OK.
232
00:13:43,870 --> 00:13:47,330
So now how come?
233
00:13:47,330 --> 00:13:48,250
How come?
234
00:13:48,250 --> 00:13:51,190
Can we even see it in
this two by two case?
235
00:13:51,190 --> 00:13:54,460
Why did -- well, I mean,
I guess we, you know,
236
00:13:54,460 --> 00:13:57,352
in one way we certainly do see
it, because it's right here.
237
00:13:57,352 --> 00:13:59,310
I mean, do we just do
it, and then we get zero.
238
00:13:59,310 --> 00:14:01,870
But we want to
think of some reason
239
00:14:01,870 --> 00:14:07,280
why the answer's zero, some
reason that we can use in the n
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00:14:07,280 --> 00:14:08,820
by n case.
241
00:14:08,820 --> 00:14:10,590
So let -- here,
here is my thinking.
242
00:14:13,450 --> 00:14:16,310
We must be, if we're
getting the answer's zero,
243
00:14:16,310 --> 00:14:20,330
we suspect that what
we're doing somehow,
244
00:14:20,330 --> 00:14:24,020
we're taking the determinant
of some matrix that
245
00:14:24,020 --> 00:14:26,660
has two equal rows.
246
00:14:26,660 --> 00:14:31,040
So I believe that if we multiply
these by the cofactors from
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00:14:31,040 --> 00:14:35,770
some other row, we're taking
the determinant -- ye,
248
00:14:35,770 --> 00:14:38,660
what matrix are we taking
the determinant of?
249
00:14:38,660 --> 00:14:40,380
Here it's, this is it.
250
00:14:40,380 --> 00:14:44,900
We're, when we do that times
this, we're really taking --
251
00:14:44,900 --> 00:14:48,510
can I put this in little
letters down here?
252
00:14:48,510 --> 00:14:57,220
I'm taking -- let me look
at the matrix a b a b.
253
00:14:57,220 --> 00:15:03,610
Let me call that matrix
AS, A screwed up.
254
00:15:03,610 --> 00:15:04,290
OK.
255
00:15:04,290 --> 00:15:05,240
All right.
256
00:15:05,240 --> 00:15:09,260
So now that matrix is
certainly singular.
257
00:15:09,260 --> 00:15:12,380
So if we find its determinant,
we're going to get zero.
258
00:15:12,380 --> 00:15:16,810
But I claim that if we find
its determinant by the cofactor
259
00:15:16,810 --> 00:15:19,960
rule, go along the
first row, we would
260
00:15:19,960 --> 00:15:23,810
take a times the cofactor of a.
261
00:15:23,810 --> 00:15:27,180
And what is the --
262
00:15:27,180 --> 00:15:32,440
see, how -- oh no -- let
me go along the second row.
263
00:15:32,440 --> 00:15:33,810
OK.
264
00:15:33,810 --> 00:15:36,320
So let's see, which --
265
00:15:36,320 --> 00:15:37,190
if I take --
266
00:15:37,190 --> 00:15:40,510
I know I've got a
singular matrix here.
267
00:15:40,510 --> 00:15:46,310
And I believe that when I
do this multiplication, what
268
00:15:46,310 --> 00:15:51,110
I'm doing is using the cofactor
formula for the determinant.
269
00:15:51,110 --> 00:15:52,700
And I know I'm
going to get zero.
270
00:15:52,700 --> 00:15:54,530
Let me try this again.
271
00:15:54,530 --> 00:15:56,920
So the cofactor formula
for the determinant
272
00:15:56,920 --> 00:16:04,170
says I should take a times
its cofactor, which is this b,
273
00:16:04,170 --> 00:16:09,260
plus b times its cofactor,
which is this minus a.
274
00:16:09,260 --> 00:16:10,040
OK.
275
00:16:10,040 --> 00:16:15,260
That's what we're doing,
apart from a sign here.
276
00:16:15,260 --> 00:16:20,980
Oh yeah, so you know, there
might be a minus multiplying
277
00:16:20,980 --> 00:16:21,950
everything.
278
00:16:21,950 --> 00:16:24,280
So if I take this
determinant, it's A --
279
00:16:24,280 --> 00:16:29,000
the determinant of this, the
determinant of A screwed up is
280
00:16:29,000 --> 00:16:33,050
a times its
cofactor, which is b,
281
00:16:33,050 --> 00:16:38,980
plus the second guy times its
cofactor, which is minus a.
282
00:16:38,980 --> 00:16:41,790
And of course I get
the answer zero,
283
00:16:41,790 --> 00:16:45,790
and this is exactly what's
happening in that, in that,
284
00:16:45,790 --> 00:16:47,650
row times this wrong column.
285
00:16:47,650 --> 00:16:48,150
OK.
286
00:16:52,740 --> 00:16:56,460
That's the two by two picture,
and it's just the same here.
287
00:16:56,460 --> 00:17:01,870
That the reason I get a zero
up in there is, the reason
288
00:17:01,870 --> 00:17:09,319
I get a zero is that when I
multiply the first row of A
289
00:17:09,319 --> 00:17:12,880
and the last row of
the cofactor matrix,
290
00:17:12,880 --> 00:17:16,450
it's as if I'm taking the
determinant of this screwed up
291
00:17:16,450 --> 00:17:19,420
matrix that has first
and last rows identical.
292
00:17:22,270 --> 00:17:26,150
The book pins this
down more specific --
293
00:17:26,150 --> 00:17:29,410
and more carefully than
I can do in the lecture.
294
00:17:29,410 --> 00:17:31,200
I hope you're seeing the point.
295
00:17:31,200 --> 00:17:34,130
That this is an identity.
296
00:17:34,130 --> 00:17:37,310
That it's a beautiful
identity and it tells us what
297
00:17:37,310 --> 00:17:40,690
the inverse of the matrix is.
298
00:17:40,690 --> 00:17:43,820
So it gives us the inverse,
the formula for the inverse.
299
00:17:43,820 --> 00:17:44,510
OK.
300
00:17:44,510 --> 00:17:49,180
So that's the first goal of
my lecture, was to find this
301
00:17:49,180 --> 00:17:49,960
formula.
302
00:17:49,960 --> 00:17:51,490
It's done.
303
00:17:51,490 --> 00:17:53,800
OK.
304
00:17:53,800 --> 00:17:58,180
And of course I could
invert, now, I can,
305
00:17:58,180 --> 00:18:00,900
I sort of like I can see what --
306
00:18:00,900 --> 00:18:02,850
I can answer
questions like this.
307
00:18:02,850 --> 00:18:08,510
Suppose I have a matrix,
and let me move the one one
308
00:18:08,510 --> 00:18:09,740
entry.
309
00:18:09,740 --> 00:18:13,600
What happens to the inverse?
310
00:18:13,600 --> 00:18:15,780
Just, just think
about that question.
311
00:18:15,780 --> 00:18:18,030
Suppose I have some
matrix, I just write down
312
00:18:18,030 --> 00:18:21,300
some nice, non-singular
matrix that's got an inverse,
313
00:18:21,300 --> 00:18:25,170
and then I move the one
one entry a little bit.
314
00:18:25,170 --> 00:18:27,880
I add one to it, for example.
315
00:18:27,880 --> 00:18:30,730
What happens to
the inverse matrix?
316
00:18:30,730 --> 00:18:33,160
Well, this formula
should tell me.
317
00:18:33,160 --> 00:18:36,110
I have to look to see what
happens to the determinant
318
00:18:36,110 --> 00:18:39,400
and what happens to
all the cofactors.
319
00:18:39,400 --> 00:18:44,300
And, the picture,
it's all there.
320
00:18:44,300 --> 00:18:44,960
It's all there.
321
00:18:44,960 --> 00:18:49,420
We can really understand
how the inverse changes
322
00:18:49,420 --> 00:18:51,250
when the matrix changes.
323
00:18:51,250 --> 00:18:52,330
OK.
324
00:18:52,330 --> 00:18:58,910
Now my second application is to
-- let me put that over here --
325
00:18:58,910 --> 00:18:59,630
is to Ax=b.
326
00:19:03,650 --> 00:19:09,060
Well, the -- course, the
solution is A inverse b.
327
00:19:09,060 --> 00:19:11,820
But now I have a
formula for A inverse.
328
00:19:11,820 --> 00:19:17,170
A inverse is one
over the determinant
329
00:19:17,170 --> 00:19:22,030
times this C transpose times B.
330
00:19:22,030 --> 00:19:24,510
I now know what A inverse is.
331
00:19:24,510 --> 00:19:27,270
So now I just have to
say, what have I got here?
332
00:19:27,270 --> 00:19:33,600
Is there any way to, to make
this formula, this answer,
333
00:19:33,600 --> 00:19:36,550
which is the one
and only answer --
334
00:19:36,550 --> 00:19:40,980
it's the very same answer we got
on the first day of the class
335
00:19:40,980 --> 00:19:42,620
by elimination.
336
00:19:42,620 --> 00:19:47,290
Now I'm -- now I've got
a formula for the answer.
337
00:19:47,290 --> 00:19:51,890
Can I play with it further
to see what's going on?
338
00:19:51,890 --> 00:20:01,640
And Cramer's, this Cramer's
Rule is exactly, that --
339
00:20:01,640 --> 00:20:07,010
a way of looking
at this formula.
340
00:20:07,010 --> 00:20:07,510
OK.
341
00:20:07,510 --> 00:20:11,620
So this is a formula for x.
342
00:20:11,620 --> 00:20:13,620
Here's my formula.
343
00:20:13,620 --> 00:20:14,350
Well, of course.
344
00:20:14,350 --> 00:20:16,050
The first thing I
see from the formula
345
00:20:16,050 --> 00:20:21,100
is that the answer x always
has that in the determinant.
346
00:20:21,100 --> 00:20:22,590
I'm not surprised.
347
00:20:22,590 --> 00:20:25,240
There's a division
by the determinant.
348
00:20:25,240 --> 00:20:29,190
But then I have to say a little
more carefully what's going on
349
00:20:29,190 --> 00:20:31,610
And let me tell you what
Cramer's Rule is. up here.
350
00:20:31,610 --> 00:20:35,610
Let, let me take x1,
the first component.
351
00:20:35,610 --> 00:20:37,740
So this is the first
component of the answer.
352
00:20:37,740 --> 00:20:40,870
Then there'll be a
second component and a,
353
00:20:40,870 --> 00:20:42,900
all the other components.
354
00:20:42,900 --> 00:20:46,310
Can I take just the first
component of this formula?
355
00:20:46,310 --> 00:20:52,040
Well, I certainly have
determinant of A down under.
356
00:20:52,040 --> 00:20:56,400
And what the heck
is the first --
357
00:20:56,400 --> 00:21:00,300
so what do I get
in C transpose b?
358
00:21:00,300 --> 00:21:03,440
What's the first entry
of C transpose b?
359
00:21:03,440 --> 00:21:06,060
That's what I have
to answer myself.
360
00:21:06,060 --> 00:21:10,420
Well, what's the first
entry of C transpose b?
361
00:21:13,820 --> 00:21:17,670
This B is -- let me
tell you what it is.
362
00:21:17,670 --> 00:21:18,170
OK.
363
00:21:18,170 --> 00:21:22,290
Somehow I'm
multiplying cofactors
364
00:21:22,290 --> 00:21:26,970
by the entries of B,
right, in this product.
365
00:21:26,970 --> 00:21:30,430
Cofactors from the matrix
times entries of b.
366
00:21:30,430 --> 00:21:33,650
So any time I'm multiplying
cofactors by numbers,
367
00:21:33,650 --> 00:21:36,980
I think, I'm getting the
determinant of something.
368
00:21:36,980 --> 00:21:39,990
And let me call
that something B1.
369
00:21:39,990 --> 00:21:46,730
So this is a matrix, the matrix
whose determinant is coming out
370
00:21:46,730 --> 00:21:47,230
of that.
371
00:21:47,230 --> 00:21:49,530
And we'll, we'll see what it is.
372
00:21:49,530 --> 00:21:54,550
x2 will be the determinant
of some other matrix B2, also
373
00:21:54,550 --> 00:21:57,650
divided by determinant of A.
374
00:21:57,650 --> 00:21:59,190
So now I just --
375
00:21:59,190 --> 00:22:00,970
Cramer just had a good idea.
376
00:22:00,970 --> 00:22:06,920
He realized what matrix it was,
what these B1 and B2 and B3
377
00:22:06,920 --> 00:22:08,470
and so on matrices were.
378
00:22:08,470 --> 00:22:10,520
Let me write them on
the board underneath.
379
00:22:14,610 --> 00:22:15,260
OK.
380
00:22:15,260 --> 00:22:17,470
So what is this B1?
381
00:22:17,470 --> 00:22:24,430
This B1 is the matrix that
has b in its first column
382
00:22:24,430 --> 00:22:27,900
and otherwise the
rest of it is A.
383
00:22:27,900 --> 00:22:38,740
So it otherwise it has the
rest, the, the n-1 columns of A.
384
00:22:38,740 --> 00:22:42,880
It's the matrix with --
385
00:22:42,880 --> 00:22:51,410
it's just the matrix A
with column one replaced
386
00:22:51,410 --> 00:22:58,520
by the right-hand side,
by the right-hand side b.
387
00:22:58,520 --> 00:23:04,490
Because somehow when I take
the determinant of this guy,
388
00:23:04,490 --> 00:23:09,140
it's giving me this
matrix multiplication.
389
00:23:09,140 --> 00:23:10,570
Well, how could that be?
390
00:23:13,180 --> 00:23:16,490
How could -- so what's,
what's the determinant formula
391
00:23:16,490 --> 00:23:18,050
I'll use here?
392
00:23:18,050 --> 00:23:21,910
I'll use cofactors, of course.
393
00:23:21,910 --> 00:23:25,240
And I might as well use
cofactors down column one.
394
00:23:25,240 --> 00:23:27,910
So when I use cofactors
down column one,
395
00:23:27,910 --> 00:23:32,840
I'm taking the first
entry of b times what?
396
00:23:32,840 --> 00:23:35,650
Times the cofactor c11.
397
00:23:35,650 --> 00:23:38,410
Do you see that?
398
00:23:38,410 --> 00:23:40,830
When I, when I use
cofactors here,
399
00:23:40,830 --> 00:23:43,530
I take the first
entry here, B one
400
00:23:43,530 --> 00:23:47,120
let's call it, times
the cofactor there.
401
00:23:47,120 --> 00:23:51,390
But what's the cofactor in --
my little hand-waving is meant
402
00:23:51,390 --> 00:23:54,650
to indicate that it's a
matrix of one size smaller,
403
00:23:54,650 --> 00:23:56,030
the cofactor.
404
00:23:56,030 --> 00:23:58,730
And it's exactly c11.
405
00:23:58,730 --> 00:24:01,130
Well, that's just
what we wanted.
406
00:24:01,130 --> 00:24:06,200
This first entry
is c11 times b1.
407
00:24:06,200 --> 00:24:13,970
And then the next entry is
whatever, is c21 times b2,
408
00:24:13,970 --> 00:24:14,860
and so on.
409
00:24:14,860 --> 00:24:17,310
And sure enough, if
I look here, when
410
00:24:17,310 --> 00:24:19,720
I'm, when I do the
cofactor expansion,
411
00:24:19,720 --> 00:24:22,790
b2 is getting multiplied by
the right thing, and so on.
412
00:24:25,450 --> 00:24:27,400
So there's Cramer's Rule.
413
00:24:27,400 --> 00:24:31,560
And the book gives
another kind of cute proof
414
00:24:31,560 --> 00:24:36,960
without, without building
so much on, on cofactors.
415
00:24:36,960 --> 00:24:38,950
But here we've got
cofactors, so I thought
416
00:24:38,950 --> 00:24:40,650
I'd just give you this proof.
417
00:24:40,650 --> 00:24:42,700
So what is B --
418
00:24:42,700 --> 00:24:45,430
in general, what is Bj?
419
00:24:45,430 --> 00:24:58,970
This is the, this is A with
column j replaced by, by b.
420
00:25:02,920 --> 00:25:08,870
So that's -- the determinant
of that matrix that you divide
421
00:25:08,870 --> 00:25:11,700
by determinant of A to get xj.
422
00:25:11,700 --> 00:25:14,980
So x -- let me change
this general formula.
423
00:25:14,980 --> 00:25:18,190
xj, the j-th one,
is the determinant
424
00:25:18,190 --> 00:25:22,510
of Bj divided by the
determinant of A.
425
00:25:22,510 --> 00:25:24,360
And now we've said what Bj is.
426
00:25:30,160 --> 00:25:33,720
Well, so Cramer found a rule.
427
00:25:33,720 --> 00:25:39,320
And we could ask him, OK,
great, good work, Cramer.
428
00:25:39,320 --> 00:25:44,180
But is your rule any
good in practice?
429
00:25:44,180 --> 00:25:49,960
So he says, well, you couldn't
ask about a rule in mine,
430
00:25:49,960 --> 00:25:52,220
right, because it's just --
431
00:25:52,220 --> 00:25:56,850
all you have to do is find
the determinant of A and these
432
00:25:56,850 --> 00:26:01,220
other determinants, so I guess
-- oh, he just says, well,
433
00:26:01,220 --> 00:26:04,140
all you have to do is
find n+1 determinants,
434
00:26:04,140 --> 00:26:06,170
the, the n Bs and the A.
435
00:26:06,170 --> 00:26:17,530
And actually, I remember
reading -- there was a book,
436
00:26:17,530 --> 00:26:22,530
popular book that, that kids
interested in math read when I
437
00:26:22,530 --> 00:26:25,980
was a kid interested in
math called Mathematics
438
00:26:25,980 --> 00:26:29,840
for the Million or something,
by a guy named Bell.
439
00:26:29,840 --> 00:26:34,830
And it had a little page
about linear algebra.
440
00:26:34,830 --> 00:26:39,000
And it said,-- so it
explained elimination
441
00:26:39,000 --> 00:26:41,250
in a very complicated way.
442
00:26:41,250 --> 00:26:43,200
I certainly didn't
understand it.
443
00:26:43,200 --> 00:26:47,950
And, and it made it, you
know, it sort of said, well,
444
00:26:47,950 --> 00:26:51,120
there is this formula
for elimination,
445
00:26:51,120 --> 00:26:55,770
but look at this great
formula, Cramer's Rule.
446
00:26:55,770 --> 00:27:00,410
So it certainly said Cramer's
Rule was the way to go.
447
00:27:00,410 --> 00:27:05,180
But actually, Cramer's Rule
is a disastrous way to go,
448
00:27:05,180 --> 00:27:07,920
because to compute
these determinants,
449
00:27:07,920 --> 00:27:12,060
it takes, like,
approximately forever.
450
00:27:12,060 --> 00:27:17,020
So actually I now think
of that book title
451
00:27:17,020 --> 00:27:18,980
as being Mathematics
for the Millionaire,
452
00:27:18,980 --> 00:27:22,160
because you'd have to
be able to pay for,
453
00:27:22,160 --> 00:27:26,570
a hopelessly long calculation
where elimination, of course,
454
00:27:26,570 --> 00:27:30,450
produced the x-s, in an instant.
455
00:27:30,450 --> 00:27:36,060
But having a formula allows
you to, with, with letters, you
456
00:27:36,060 --> 00:27:40,010
know, allows you to do algebra
instead of, algorithms.
457
00:27:40,010 --> 00:27:44,220
So the, there's some value
in the Cramer's Rule formula
458
00:27:44,220 --> 00:27:52,000
for x and in the explicit
formula for, for A inverse.
459
00:27:52,000 --> 00:27:54,890
They're nice
formulas, but I just
460
00:27:54,890 --> 00:27:57,710
don't want you to use them.
461
00:27:57,710 --> 00:27:59,230
That'ss what it comes to.
462
00:27:59,230 --> 00:28:02,960
If you had to -- and Matlab
would never, never do it.
463
00:28:02,960 --> 00:28:05,480
I mean, it would
use elimination.
464
00:28:05,480 --> 00:28:06,720
OK.
465
00:28:06,720 --> 00:28:11,380
Now I'm ready for number
three in today's list
466
00:28:11,380 --> 00:28:17,790
of amazing connections coming
through the determinant.
467
00:28:17,790 --> 00:28:21,900
And that number three is the
fact that the determinant gives
468
00:28:21,900 --> 00:28:22,950
a volume.
469
00:28:22,950 --> 00:28:24,480
OK.
470
00:28:24,480 --> 00:28:28,860
So now -- so that's
my final topic for --
471
00:28:28,860 --> 00:28:33,070
among these -- this my
number three application,
472
00:28:33,070 --> 00:28:37,700
that the determinant is actually
equals the volume of something.
473
00:28:37,700 --> 00:28:42,720
Can I use this little space
to consider a special case,
474
00:28:42,720 --> 00:28:46,050
and then I'll use the
far board to think
475
00:28:46,050 --> 00:28:47,840
about the general rule.
476
00:28:47,840 --> 00:28:50,400
So what I going to prove?
477
00:28:50,400 --> 00:28:51,970
Or claim.
478
00:28:51,970 --> 00:28:56,950
I claim that the
determinant of the matrix
479
00:28:56,950 --> 00:29:02,800
is the volume of a box.
480
00:29:02,800 --> 00:29:05,930
OK, and you say, which box?
481
00:29:05,930 --> 00:29:06,970
Fair enough.
482
00:29:06,970 --> 00:29:08,570
OK.
483
00:29:08,570 --> 00:29:11,670
So let's see.
484
00:29:11,670 --> 00:29:16,170
I'm in -- shall we say we're
in, say three by three?
485
00:29:16,170 --> 00:29:18,780
Shall we suppose -- let's,
let's say three by three.
486
00:29:18,780 --> 00:29:22,680
So, so we can really -- we're,
we're talking about boxes
487
00:29:22,680 --> 00:29:26,400
in three dimensions, and
three by three matrices.
488
00:29:26,400 --> 00:29:30,990
And so all I do -- you
could guess what the box is.
489
00:29:30,990 --> 00:29:34,700
Here is, here is,
three dimensions.
490
00:29:34,700 --> 00:29:35,450
OK.
491
00:29:35,450 --> 00:29:41,880
Now I take the first row of
the matrix, a11, a22, A --
492
00:29:41,880 --> 00:29:44,360
sorry. a11, a12, a13.
493
00:29:44,360 --> 00:29:48,090
That row is a vector.
494
00:29:48,090 --> 00:29:50,190
It goes to some point.
495
00:29:50,190 --> 00:29:53,220
That point will be -- and
that edge going to it,
496
00:29:53,220 --> 00:29:56,650
will be an edge of the box,
and that point will be a corner
497
00:29:56,650 --> 00:29:57,740
of the box.
498
00:29:57,740 --> 00:30:01,010
So here is zero zero
zero, of course.
499
00:30:01,010 --> 00:30:09,500
And here's the first row of
the matrix: a11, a12, a13.
500
00:30:09,500 --> 00:30:14,840
So that's one edge of the box.
501
00:30:14,840 --> 00:30:19,700
Another edge of the box
is to the second row
502
00:30:19,700 --> 00:30:21,820
of the matrix, row two.
503
00:30:21,820 --> 00:30:24,510
Can I just call
it there row two?
504
00:30:24,510 --> 00:30:28,760
And a third row of
the box will be to --
505
00:30:28,760 --> 00:30:32,200
a third row -- a third edge of
the box will be given by row
506
00:30:32,200 --> 00:30:33,300
three.
507
00:30:33,300 --> 00:30:36,470
So, so there's row three.
508
00:30:36,470 --> 00:30:38,780
That, the coordinates,
what are the coordinates
509
00:30:38,780 --> 00:30:41,030
of that corner of the box?
510
00:30:41,030 --> 00:30:47,650
a31, a32, a33.
511
00:30:47,650 --> 00:30:51,220
So I've got that edge of the
box, that edge of the box,
512
00:30:51,220 --> 00:30:53,830
that edge of the box,
and that's all I need.
513
00:30:53,830 --> 00:30:59,820
Now I just finish
out the box, right?
514
00:30:59,820 --> 00:31:03,650
I just -- the proper word,
of course, is parallelepiped.
515
00:31:03,650 --> 00:31:08,920
But for obvious
reasons, I wrote box.
516
00:31:08,920 --> 00:31:09,550
OK.
517
00:31:09,550 --> 00:31:11,360
So, OK.
518
00:31:11,360 --> 00:31:14,550
So there's the, there's
the bottom of the box.
519
00:31:14,550 --> 00:31:19,660
There're the four
edge sides of the box.
520
00:31:19,660 --> 00:31:23,570
There's the top of the box.
521
00:31:23,570 --> 00:31:24,540
Cute, right?
522
00:31:24,540 --> 00:31:28,470
It's the box that
has these three edges
523
00:31:28,470 --> 00:31:33,080
and then it's completed to
a, to a, each, you know,
524
00:31:33,080 --> 00:31:35,080
each side is a, is
a parallelogram.
525
00:31:37,730 --> 00:31:43,050
And it's that box whose volume
is given by the determinant.
526
00:31:45,610 --> 00:31:50,650
That's -- now it's -- possible
that the determinant is
527
00:31:50,650 --> 00:31:52,920
negative.
528
00:31:52,920 --> 00:31:56,110
So we have to just say
what's going on in that case.
529
00:31:56,110 --> 00:32:01,840
If the determinant is
negative, then the volume, we,
530
00:32:01,840 --> 00:32:04,030
we should take the
absolute value really.
531
00:32:04,030 --> 00:32:07,030
So the volume, if we, if we
think of volume as positive,
532
00:32:07,030 --> 00:32:11,310
we should take the absolute
value of the determinant.
533
00:32:11,310 --> 00:32:14,400
But the, the sign, what does
the sign of the determinant --
534
00:32:14,400 --> 00:32:16,560
it always must
tell us something.
535
00:32:16,560 --> 00:32:20,380
And somehow it, it will tell
us whether these three is a --
536
00:32:20,380 --> 00:32:23,270
whether it's a right-handed
box or a left-handed box.
537
00:32:23,270 --> 00:32:30,300
If we, if we reversed
two of the edges,
538
00:32:30,300 --> 00:32:32,250
we would go between
a right-handed box
539
00:32:32,250 --> 00:32:33,390
and a left-handed box.
540
00:32:33,390 --> 00:32:35,440
We wouldn't change the
volume, but we would
541
00:32:35,440 --> 00:32:40,500
change the, the cyclic, order.
542
00:32:40,500 --> 00:32:42,900
So I won't worry about that.
543
00:32:42,900 --> 00:32:47,020
And, so one special
case is what?
544
00:32:47,020 --> 00:32:49,930
A equal identity matrix.
545
00:32:49,930 --> 00:32:52,070
So let's take that special case.
546
00:32:52,070 --> 00:32:55,520
A equal identity matrix.
547
00:32:55,520 --> 00:32:59,550
Is the formula determinant
of identity matrix,
548
00:32:59,550 --> 00:33:01,230
does that equal the
volume of the box?
549
00:33:04,300 --> 00:33:06,460
Well, what is the box?
550
00:33:06,460 --> 00:33:07,370
What's the box?
551
00:33:07,370 --> 00:33:12,230
If A is the identity matrix,
then these three rows are
552
00:33:12,230 --> 00:33:17,460
the three coordinate
vectors, and the box is --
553
00:33:17,460 --> 00:33:18,920
it's a cube.
554
00:33:18,920 --> 00:33:21,200
It's the unit cube.
555
00:33:21,200 --> 00:33:23,970
So if, if A is the
identity matrix, of course
556
00:33:23,970 --> 00:33:26,080
our formula is
557
00:33:26,080 --> 00:33:29,850
Well, actually that
proves property one --
558
00:33:29,850 --> 00:33:31,740
that the volume right.
has property one.
559
00:33:31,740 --> 00:33:35,440
Actually, we could, we could,
we could get this thing if we --
560
00:33:35,440 --> 00:33:39,210
if we can show that the box
volume has the same three
561
00:33:39,210 --> 00:33:42,010
properties that define
the determinant,
562
00:33:42,010 --> 00:33:44,060
then it must be the determinant.
563
00:33:46,670 --> 00:33:51,080
So that's like the, the, the
elegant way to prove this.
564
00:33:51,080 --> 00:33:53,860
To prove this amazing fact
that the determinant equals
565
00:33:53,860 --> 00:33:58,540
the volume, first we'll check
it for the identity matrix.
566
00:33:58,540 --> 00:34:00,470
That's fine.
567
00:34:00,470 --> 00:34:03,190
The box is a cube
and its volume is one
568
00:34:03,190 --> 00:34:07,180
and the determinant is one
and, and one agrees with one.
569
00:34:07,180 --> 00:34:11,590
Now let me take one -- let me
go up one level to an orthogonal
570
00:34:11,590 --> 00:34:12,889
matrix.
571
00:34:12,889 --> 00:34:16,060
Because I'd like to take this
chance to bring in chapter --
572
00:34:16,060 --> 00:34:18,750
the, the previous chapter.
573
00:34:18,750 --> 00:34:21,100
Suppose I have an
orthogonal matrix.
574
00:34:21,100 --> 00:34:22,110
What did that mean?
575
00:34:22,110 --> 00:34:25,500
I always called those things Q.
576
00:34:25,500 --> 00:34:27,909
What was the point
of -- suppose I have,
577
00:34:27,909 --> 00:34:32,670
suppose instead of the identity
matrix I'm now going to take A
578
00:34:32,670 --> 00:34:36,910
equal Q, an orthogonal matrix.
579
00:34:42,940 --> 00:34:46,510
What was Q then?
580
00:34:46,510 --> 00:34:52,179
That was a matrix whose columns
were orthonormal, right?
581
00:34:52,179 --> 00:34:54,610
Those were its columns
were unit vectors,
582
00:34:54,610 --> 00:34:56,059
perpendicular unit vectors.
583
00:34:59,220 --> 00:35:03,090
So what kind of a
box have we got now?
584
00:35:03,090 --> 00:35:06,520
What kind of a box comes
from the rows or the columns,
585
00:35:06,520 --> 00:35:08,530
I don't mind, because
the determinant
586
00:35:08,530 --> 00:35:10,420
is the determinant
of the transpose,
587
00:35:10,420 --> 00:35:11,850
so I'm never worried about
588
00:35:11,850 --> 00:35:12,730
that.
589
00:35:12,730 --> 00:35:15,850
What kind of a box,
what shape box have we
590
00:35:15,850 --> 00:35:20,380
got if the matrix is
an orthogonal matrix?
591
00:35:20,380 --> 00:35:23,040
It's another cube.
592
00:35:23,040 --> 00:35:24,710
It's a cube again.
593
00:35:24,710 --> 00:35:27,900
How is it different
from the identity cube?
594
00:35:30,510 --> 00:35:33,140
It's just rotated.
595
00:35:33,140 --> 00:35:36,830
It's just the orthogonal
matrix Q doesn't
596
00:35:36,830 --> 00:35:38,860
have to be the identity matrix.
597
00:35:38,860 --> 00:35:43,240
It's just the unit cube
but turned in space.
598
00:35:43,240 --> 00:35:48,490
So sure enough, it's the unit
cube, and its volume is one.
599
00:35:48,490 --> 00:35:53,410
Now is the determinant one?
600
00:35:53,410 --> 00:35:55,410
What's the determinant of Q?
601
00:35:55,410 --> 00:35:58,980
We believe that the determinant
of Q better be one or minus
602
00:35:58,980 --> 00:36:03,100
one, so that our formula
is -- checks out in that --
603
00:36:03,100 --> 00:36:06,800
if we can't check it in these
easy cases where we got a cube,
604
00:36:06,800 --> 00:36:11,020
we're not going to get
it in the general case.
605
00:36:11,020 --> 00:36:18,290
So why is the determinant
of Q equal one or minus one?
606
00:36:18,290 --> 00:36:19,810
What do we know about Q?
607
00:36:19,810 --> 00:36:25,050
What's the one matrix statement
of the properties of Q?
608
00:36:25,050 --> 00:36:29,020
A matrix with orthonormal
columns has --
609
00:36:29,020 --> 00:36:31,860
satisfies a certain equation.
610
00:36:31,860 --> 00:36:33,400
What, what is that?
611
00:36:33,400 --> 00:36:38,930
It's if we have this orthogonal
matrix, then the fact --
612
00:36:38,930 --> 00:36:45,550
the way to say what it, what
its properties are is this.
613
00:36:45,550 --> 00:36:50,940
Q prime, u- u- Q
transpose Q equals I.
614
00:36:50,940 --> 00:36:52,790
Right?
615
00:36:52,790 --> 00:36:57,450
That's what -- those are the
matrices that get the name Q,
616
00:36:57,450 --> 00:37:01,241
the matrices that
Q transpose Q is I.
617
00:37:01,241 --> 00:37:01,740
OK.
618
00:37:01,740 --> 00:37:07,180
Now from that, tell me
why is the determinant one
619
00:37:07,180 --> 00:37:09,500
or minus one.
620
00:37:09,500 --> 00:37:12,930
How do I, out of this fact --
621
00:37:12,930 --> 00:37:14,525
this may even be a
homework problem.
622
00:37:17,240 --> 00:37:20,890
It's there in the, in the
list of exercises in the book,
623
00:37:20,890 --> 00:37:23,100
and let's just do it.
624
00:37:23,100 --> 00:37:27,340
How do I get, how do I discover
that the determinant of Q
625
00:37:27,340 --> 00:37:31,530
is one or maybe minus one?
626
00:37:31,530 --> 00:37:34,180
I take determinants of
both sides, everybody says,
627
00:37:34,180 --> 00:37:36,870
so I won't --
628
00:37:36,870 --> 00:37:38,610
I take determinants
of both sides.
629
00:37:38,610 --> 00:37:41,250
On the right-hand side -- so
I, when I take determinants
630
00:37:41,250 --> 00:37:46,240
of both sides,
let me just do it.
631
00:37:46,240 --> 00:37:48,315
Take the determinant of
-- take determinants.
632
00:37:51,680 --> 00:37:53,970
Determinant of the
identity is one.
633
00:37:53,970 --> 00:37:55,840
What's the determinant
of that product?
634
00:37:58,690 --> 00:38:02,400
Rule nine is paying off now.
635
00:38:02,400 --> 00:38:07,920
The determinant of a product is
the determinant of this guy --
636
00:38:07,920 --> 00:38:11,110
maybe I'll put it, I'll use
that symbol for determinant.
637
00:38:11,110 --> 00:38:14,152
It's the determinant of that
guy times the determinant
638
00:38:14,152 --> 00:38:14,860
of the other guy.
639
00:38:17,460 --> 00:38:21,400
And then what's the
determinant of Q transpose?
640
00:38:21,400 --> 00:38:22,990
It's the same as the
determinant of Q.
641
00:38:22,990 --> 00:38:24,810
Rule ten pays off.
642
00:38:24,810 --> 00:38:28,330
So this is just
this thing squared.
643
00:38:28,330 --> 00:38:33,740
So that determinant squared is
one and sure enough it's one
644
00:38:33,740 --> 00:38:35,290
or minus one.
645
00:38:35,290 --> 00:38:35,790
Great.
646
00:38:38,370 --> 00:38:42,000
So in these special
cases of cubes,
647
00:38:42,000 --> 00:38:48,650
we really do have
determinant equals volume.
648
00:38:48,650 --> 00:38:54,530
Now can I just push
that to non-cubes.
649
00:38:54,530 --> 00:39:02,930
Let me push it first to
rectangles, rectangular boxes,
650
00:39:02,930 --> 00:39:07,820
where I'm just multiplying
the e- the edges are --
651
00:39:07,820 --> 00:39:10,440
let me keep all the
ninety degree angles,
652
00:39:10,440 --> 00:39:13,510
because those are -- that,
that makes my life easy.
653
00:39:13,510 --> 00:39:16,910
And just stretch the edges.
654
00:39:16,910 --> 00:39:21,910
Suppose I stretch that first
edge, suppose this first edge
655
00:39:21,910 --> 00:39:23,230
I double.
656
00:39:23,230 --> 00:39:27,990
Suppose I double
that first edge,
657
00:39:27,990 --> 00:39:31,060
keeping the other
edges the same.
658
00:39:31,060 --> 00:39:34,650
What happens to the volume?
659
00:39:34,650 --> 00:39:36,640
It doubles, right?
660
00:39:36,640 --> 00:39:39,420
We know that the volume
of a cube doubles.
661
00:39:39,420 --> 00:39:42,370
In fact, because we know that
the new cube would sit right
662
00:39:42,370 --> 00:39:43,400
on top --
663
00:39:43,400 --> 00:39:46,210
I mean, the new, the added
cube would sit right on --
664
00:39:46,210 --> 00:39:47,620
would fit --
665
00:39:47,620 --> 00:39:50,120
probably a geometer would
say congruent or something --
666
00:39:50,120 --> 00:39:51,810
would go right in, in the other.
667
00:39:51,810 --> 00:39:52,830
We'd have two.
668
00:39:52,830 --> 00:39:54,540
We have two identical cubes.
669
00:39:54,540 --> 00:39:58,421
Total volume is now two.
670
00:39:58,421 --> 00:39:58,920
OK.
671
00:39:58,920 --> 00:40:02,392
So I want -- if I double an
edge, the volume doubles.
672
00:40:02,392 --> 00:40:03,725
What happens to the determinant?
673
00:40:07,580 --> 00:40:14,880
If I double, the first
row of a matrix, what ch-
674
00:40:14,880 --> 00:40:18,010
ch- what's the effect
on the determinant?
675
00:40:18,010 --> 00:40:21,320
It also doubles, right?
676
00:40:21,320 --> 00:40:25,910
And that was rule number 3a.
677
00:40:25,910 --> 00:40:29,500
Remember rule 3a was
that if I, I could,
678
00:40:29,500 --> 00:40:36,780
if I had a factor in, in row
one, T, I could factor it out.
679
00:40:36,780 --> 00:40:41,020
So if, if I have a factor
two in that row one,
680
00:40:41,020 --> 00:40:43,340
I can factor it out
of the determinant.
681
00:40:43,340 --> 00:40:48,860
It agrees with the -- the volume
of the box has that factor two.
682
00:40:48,860 --> 00:40:53,140
So, so volume satisfies
this property 3a.
683
00:40:53,140 --> 00:40:59,040
And now I really close, but I
-- but to get to the very end
684
00:40:59,040 --> 00:41:01,980
of this proof, I have
to get away from right
685
00:41:01,980 --> 00:41:02,650
angles.
686
00:41:02,650 --> 00:41:10,330
I have to allow the
possibility of, other angles.
687
00:41:10,330 --> 00:41:13,090
And -- or what's
saying the same thing,
688
00:41:13,090 --> 00:41:18,600
I have to check that the
volume also satisfies 3b.
689
00:41:18,600 --> 00:41:20,600
So can I --
690
00:41:20,600 --> 00:41:24,740
This is end of proof
that the -- so I'm --
691
00:41:24,740 --> 00:41:34,380
determinant of A equals volume
of box, and where I right now?
692
00:41:34,380 --> 00:41:40,820
This volume has properties,
properties one, no problem.
693
00:41:40,820 --> 00:41:44,030
If the box is the
cube, everything is --
694
00:41:44,030 --> 00:41:49,000
if the box is the unit
cube, its volume is one.
695
00:41:49,000 --> 00:41:54,680
Property two was if
I reverse two rows,
696
00:41:54,680 --> 00:41:57,960
but that doesn't change the box.
697
00:41:57,960 --> 00:42:00,520
And it doesn't change the
absolute value, so no problem
698
00:42:00,520 --> 00:42:01,200
there.
699
00:42:01,200 --> 00:42:06,890
Property 3a was if I mul-
you remember what 3a was?
700
00:42:06,890 --> 00:42:10,390
So property one was about
the identity matrix.
701
00:42:10,390 --> 00:42:12,900
Property two was about
a plus or minus sign
702
00:42:12,900 --> 00:42:14,560
that I don't care about.
703
00:42:14,560 --> 00:42:18,490
Property 3a was a
factor T in a row.
704
00:42:18,490 --> 00:42:24,080
But now I've got property
three B to deal with.
705
00:42:24,080 --> 00:42:25,520
What was property 3b?
706
00:42:25,520 --> 00:42:30,270
This is a great way to
review these, properties.
707
00:42:30,270 --> 00:42:34,860
So that 3b, the property
3b said -- let's do,
708
00:42:34,860 --> 00:42:36,910
let's do two by two.
709
00:42:36,910 --> 00:42:42,620
So said that if I
had a+a', b+b', c,
710
00:42:42,620 --> 00:42:47,590
d that this equaled what?
711
00:42:47,590 --> 00:42:49,120
So this is property 3b.
712
00:42:49,120 --> 00:42:53,840
This is the linearity
in row one by itself.
713
00:42:53,840 --> 00:42:59,060
So c d is staying the same,
and I can split this into a b
714
00:42:59,060 --> 00:43:01,340
and a' b'.
715
00:43:06,160 --> 00:43:12,890
That's property 3b, at least
in the two by two case.
716
00:43:12,890 --> 00:43:15,340
And what I --
717
00:43:15,340 --> 00:43:20,540
I wanted now to show
that the volume, which
718
00:43:20,540 --> 00:43:25,040
two, two by two, that
means area, has this,
719
00:43:25,040 --> 00:43:25,790
has this property.
720
00:43:28,920 --> 00:43:32,540
Let me just emphasize that we
have got -- we're getting --
721
00:43:32,540 --> 00:43:37,150
this is a formula, then, for
the area of a parallelogram.
722
00:43:37,150 --> 00:43:40,310
The area of this parallelogram
-- can I just draw it?
723
00:43:40,310 --> 00:43:42,160
OK, here's the, here's
the parallelogram.
724
00:43:42,160 --> 00:43:45,370
I have the row a b.
725
00:43:45,370 --> 00:43:47,000
That's the first row.
726
00:43:47,000 --> 00:43:49,260
That's the point a b.
727
00:43:49,260 --> 00:43:54,060
And I tack on c d.
728
00:43:54,060 --> 00:43:57,690
c d, coming out of here.
729
00:43:57,690 --> 00:43:59,065
And I complete
the parallelogram.
730
00:44:02,560 --> 00:44:03,750
So this is --
731
00:44:03,750 --> 00:44:08,140
well, I better
make it look right.
732
00:44:08,140 --> 00:44:12,530
It's really this one that has
coordinates c d and this has
733
00:44:12,530 --> 00:44:17,040
coordinates -- well,
whatever the sum is.
734
00:44:17,040 --> 00:44:18,820
And of course
starting at zero zero.
735
00:44:22,240 --> 00:44:26,026
So we all know,
this is a+c, b+d.
736
00:44:29,140 --> 00:44:31,120
Rather than --
737
00:44:31,120 --> 00:44:33,460
I'm pausing on that
proof for a minute
738
00:44:33,460 --> 00:44:37,320
just to going back
to our formula.
739
00:44:37,320 --> 00:44:40,650
Because I want you to see
that unlike Cramer's Rule,
740
00:44:40,650 --> 00:44:44,060
that I wasn't that
impressed by, I'm
741
00:44:44,060 --> 00:44:46,640
very impressed by this
formula for the area
742
00:44:46,640 --> 00:44:48,540
of a parallelogram.
743
00:44:48,540 --> 00:44:50,930
And what's our formula?
744
00:44:50,930 --> 00:44:54,890
What, what's the area
of that parallelogram?
745
00:44:54,890 --> 00:45:00,250
If I had asked you
that last year,
746
00:45:00,250 --> 00:45:03,950
you would have said OK,
the area of a parallelogram
747
00:45:03,950 --> 00:45:06,060
is the base times the height,
748
00:45:06,060 --> 00:45:07,190
right?
749
00:45:07,190 --> 00:45:10,890
So you would have figured
out what this base, the --
750
00:45:10,890 --> 00:45:12,850
how long that base was.
751
00:45:12,850 --> 00:45:16,182
It's like the square root
of A squared plus b squared.
752
00:45:16,182 --> 00:45:17,640
And then you would
have figured out
753
00:45:17,640 --> 00:45:20,360
how much is this
height, whatever it is.
754
00:45:20,360 --> 00:45:21,920
It's horrible.
755
00:45:21,920 --> 00:45:28,330
This, I mean, we got square
roots, and in that height
756
00:45:28,330 --> 00:45:32,040
there would be other
revolting stuff.
757
00:45:32,040 --> 00:45:35,800
But now what's the formula
that we now know for the area?
758
00:45:42,290 --> 00:45:45,740
It's the determinant
of our little matrix.
759
00:45:48,540 --> 00:45:51,050
It's just ad-bc.
760
00:45:56,990 --> 00:45:59,110
No square roots.
761
00:45:59,110 --> 00:46:03,000
Totally rememberable, because
it's exactly a formula
762
00:46:03,000 --> 00:46:06,960
that we've been studying the
whole, for three lectures.
763
00:46:06,960 --> 00:46:07,460
OK.
764
00:46:11,090 --> 00:46:13,960
That's, you know, that's
the most important point
765
00:46:13,960 --> 00:46:15,190
I'm making here.
766
00:46:15,190 --> 00:46:20,990
Is that if you know the
coordinates of a box,
767
00:46:20,990 --> 00:46:24,750
of the corners, then
you have a great formula
768
00:46:24,750 --> 00:46:28,620
for the volume,
area or volume, that
769
00:46:28,620 --> 00:46:33,550
doesn't involve any lengths
or any angles or any heights,
770
00:46:33,550 --> 00:46:37,850
but just involves the
coordinates that you've got.
771
00:46:37,850 --> 00:46:40,390
And similarly, what's the
area of this triangle?
772
00:46:40,390 --> 00:46:43,530
Suppose I chop that off
and say what about --
773
00:46:43,530 --> 00:46:45,810
because you might often be
interested in a triangle
774
00:46:45,810 --> 00:46:47,330
instead of a parallelogram.
775
00:46:47,330 --> 00:46:48,725
What's the area
of this triangle?
776
00:46:52,040 --> 00:46:53,950
Now there again,
everybody would have
777
00:46:53,950 --> 00:46:58,110
said the area of a triangle is
half the base times the height.
778
00:47:01,200 --> 00:47:04,300
And in some cases, if you know
the base that a, that's --
779
00:47:04,300 --> 00:47:06,300
and the height, that's fine.
780
00:47:06,300 --> 00:47:10,280
But here, we, what we know is
the coordinates of the corners.
781
00:47:10,280 --> 00:47:11,980
We know the vertices.
782
00:47:11,980 --> 00:47:14,690
And so what's the
area of that triangle?
783
00:47:17,830 --> 00:47:22,550
If I know these, if I know
a b, c d, and zero zero,
784
00:47:22,550 --> 00:47:25,060
what's the area?
785
00:47:25,060 --> 00:47:29,670
It's just half, so
it's just half of this.
786
00:47:29,670 --> 00:47:33,040
So this is, this
is a- a b -- a d -
787
00:47:33,040 --> 00:47:38,870
b c for the parallelogram
and one half of that,
788
00:47:38,870 --> 00:47:43,585
one half of ad-bc
for the triangle.
789
00:47:47,140 --> 00:47:50,900
So I mean, this is a totally
trivial remark, to say, well,
790
00:47:50,900 --> 00:47:52,630
divide by two.
791
00:47:52,630 --> 00:47:56,910
But it's just that you
more often see triangles,
792
00:47:56,910 --> 00:48:01,530
and you feel you know
the formula for the area
793
00:48:01,530 --> 00:48:04,950
but the good formula for
the area is this one.
794
00:48:04,950 --> 00:48:06,760
And I'm just going to --
795
00:48:06,760 --> 00:48:08,260
I'm just going to
say one more thing
796
00:48:08,260 --> 00:48:09,750
about the area of a triangle.
797
00:48:09,750 --> 00:48:11,220
It's just because
it's -- you know,
798
00:48:11,220 --> 00:48:15,760
it's so great to have a
good formula for something.
799
00:48:15,760 --> 00:48:20,990
What if our triangle did
not start at zero zero?
800
00:48:20,990 --> 00:48:25,630
What if our triangle,
what if we had this --
801
00:48:25,630 --> 00:48:28,390
what if we had -- so I'm
coming back to triangles again.
802
00:48:32,600 --> 00:48:40,320
But let me, let me put this
triangle somewhere, it's --
803
00:48:40,320 --> 00:48:43,800
I'm staying with triangles,
I'm just in two dimensions,
804
00:48:43,800 --> 00:48:53,280
but I'm going to allow you
to give me any three corners.
805
00:48:57,420 --> 00:49:01,430
And in -- those six numbers
must determine the area.
806
00:49:01,430 --> 00:49:03,750
And what's the formula?
807
00:49:03,750 --> 00:49:05,280
The area is going
to be, it's going
808
00:49:05,280 --> 00:49:09,530
to be, there'll be that
half of a parallelogram.
809
00:49:09,530 --> 00:49:13,580
I mean, basically this can't
be completely new, right?
810
00:49:13,580 --> 00:49:17,230
We've got the area when -- we,
we know the area when this is
811
00:49:17,230 --> 00:49:20,400
zero zero.
812
00:49:20,400 --> 00:49:24,810
Now we just want to lift our
sight slightly and get the area
813
00:49:24,810 --> 00:49:27,820
when all th- so let me
write down what it, what it
814
00:49:27,820 --> 00:49:29,190
comes out to be.
815
00:49:29,190 --> 00:49:37,390
It turns out that if you do
this, x1 y1 and a 1, x2 y2
816
00:49:37,390 --> 00:49:44,150
and a 1, x3 y3 and a
1, that that works.
817
00:49:44,150 --> 00:49:47,320
That the determinant
symbol, of course.
818
00:49:47,320 --> 00:49:51,640
It's just -- if I gave you
that determinant to find,
819
00:49:51,640 --> 00:49:53,890
you might subtract
this row from this.
820
00:49:53,890 --> 00:49:55,780
It would kill that one.
821
00:49:55,780 --> 00:49:59,170
Subtract this row from this,
it would kill that one.
822
00:49:59,170 --> 00:50:02,470
Then you'd have a simple
determinant to do with
823
00:50:02,470 --> 00:50:06,300
differences, and it would --
824
00:50:06,300 --> 00:50:08,960
this little
subtraction, what I did
825
00:50:08,960 --> 00:50:12,580
was equivalent to
moving the triangle
826
00:50:12,580 --> 00:50:16,550
to start at the origin.
827
00:50:16,550 --> 00:50:19,850
I did it fast,
because time is up.
828
00:50:19,850 --> 00:50:24,530
And I didn't complete
that proof of 3b.
829
00:50:24,530 --> 00:50:28,930
I'll leave -- the book has a
carefully drawn figure to show
830
00:50:28,930 --> 00:50:30,870
why that works.
831
00:50:30,870 --> 00:50:34,110
But I hope you saw
the main point is
832
00:50:34,110 --> 00:50:36,760
that for area and
volume, determinant
833
00:50:36,760 --> 00:50:39,540
gives a great formula.
834
00:50:39,540 --> 00:50:40,100
OK.
835
00:50:40,100 --> 00:50:44,410
And next lectures are
about eigenvalues,
836
00:50:44,410 --> 00:50:47,780
so we're really
into the big stuff.
837
00:50:47,780 --> 00:50:49,330
Thanks.