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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 to get a formula
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for the determinant 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
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single number.
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And that number can give us
formulas for all sorts of,
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things that we've been
calculating already without
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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 and
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n by n.
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And actually maybe you can see
what's going on from this two by
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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 where it has one over
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the determinant and,
and you remember why that makes
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good sense, because then that's
perfect as long as the
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determinant isn't zero,
and that's exactly when there
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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 times
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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, because if I strike
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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,
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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,
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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,
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the -- what I'll just -- but
why don't we just call it the
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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 is the
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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
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this one, 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 of d,
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and that's this a.
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OK.
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So there's the formula.
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But we got to think why.
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I mean, it worked in this two
by two case, but a lot of other
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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 -- so this is what I
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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 -- the determinant
is a bunch of products of three
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factors, right?
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The determinant of this
matrix'll
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involve a e i,
and b f times g,
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and c times d times h,
and minus c e g,
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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
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the inverse is?
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I throw away the row and column
containing a and I take the
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determinant of what's left,
that's the cofactor.
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And that's e i minus f h.
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Products of 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
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entries.
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And, like, we never noticed any
of this stuff when we were
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computing the inverse 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
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it, we did elimination till A
became the identity.
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And then that,
the identity suddenly was A
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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 of
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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,
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let me -- I'll say,
how can we check -- what do I
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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 -- gives
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the identity.
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So I check that A times C
transpose -- let me bring the
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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,
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then, then I've correctly
identified A inverse as C
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transpose divided 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 -- here,
here's A, here's a11 -- I'm
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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 would mean I'm in row one,
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but I've transposed,
so that's, those are the
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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 is.
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This is like you're seeing the
main point if you just tell me
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one entry.
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What do I get up there in the
one
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one entry when I do this row of
this row from A times this
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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 on what
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the last lecture reached.
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So this is a11 times c11,
a12 times c12,
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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, which I wrote,
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which we got last time.
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The determinant of A is,
if I use row one,
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let,
let I equal one,
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then I have a11 times its
cofactor, a12 times its
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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,
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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?
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That's just the cofactors,
that's just row two times its
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cofactors.
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So of course I get the
determinant
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again.
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And in the last here,
this is the last row times its
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cofactors.
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It's exactly -- you see,
we're realizing that the
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cofactor 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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Great.
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But there's one more idea here,
right?
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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
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all the off-diagonal stuff,
which I want to be zero,
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right?
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I just want this to be
determinant of A times the
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identity, and then I'm,
I'm a happy person.
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So why should that be?
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Why should it be that one row
times the cofactors from a
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different row,
which become a column because I
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transpose, give zero?
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In other words,
the cofactor formula gives the
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determinant if the
row and the,
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and the cofactors -- you know,
if the entries of A and the
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cofactors are for the same row.
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But for some reason,
if I take the cofactors from
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the -- entries from the first
row and the cofactors from the
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second row,
for some reason I automatically
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get zero.
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And it's sort of like
interesting to say,
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why does that happen?
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And can I just check that -- of
course, we know it happens,
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in this case.
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Here are the numbers from row
one and here are the cofactors
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from row two,
right?
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Those are the numbers in row
one.
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And th- these are the cofactors
from row two,
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because the cofactor of c is
minus b and the cofactor of d is
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a.
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And sure enough,
that row times this column
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gives -- please say it.
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Zero, right.
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OK.
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So now how come?
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How come?
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Can we even see it in
this two by two case?
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Why did -- well,
I mean, I guess we,
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you know, in one way we
certainly do see it,
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because it's right here.
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I mean, do we just do it,
and then we get zero.
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But we want to think of some
reason why the answer's zero,
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some reason that we can use in
the n by n case.
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So let -- here,
here is my thinking.
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We must be, if we're getting
the answer's zero,
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we suspect that what we're
doing somehow,
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we're taking the determinant of
some matrix that has two equal
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rows.
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So I believe that if we
multiply these by the cofactors
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from some other row,
we're taking the determinant --
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ye, what matrix are we taking
the determinant of?
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Here it's, this is it.
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We're, when we do that times
this, we're really taking -- can
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I put this in little letters
down here?
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I'm taking -- let me look at
the matrix a b a b.
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Let me call that matrix AS,
A screwed up.
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OK.
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All right.
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So now that matrix is certainly
singular.
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So if we find its determinant,
we're going to get zero.
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But I claim that if we find its
determinant by the cofactor
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rule,
go along the first row,
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we would take a times the
cofactor of a.
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And what is the -- see,
how -- oh no -- let me go along
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the second row.
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OK.
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So let's see,
which -- if I take -- I know
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I've got a singular matrix here.
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And I believe that when I do
this multiplication,
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what I'm doing is using the
cofactor formula for the
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determinant.
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And I know I'm going to get
zero.
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Let me try this again.
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So the cofactor formula for the
determinant says I should take a
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times its cofactor,
which is this b,
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plus b times its cofactor,
which is this minus a.
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OK.
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That's what we're doing,
apart from a sign here.
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Oh yeah, so you know,
there might be a minus
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multiplying everything.
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So if I take this determinant,
it's A -- the determinant of
283
00:16:25 --> 00:16:29
this, the determinant of A
screwed up is a times its
284
00:16:29 --> 00:16:33
cofactor, which is b,
plus the second guy
285
00:16:33 --> 00:16:37
times its cofactor,
which is minus a.
286
00:16:37 --> 00:16:43
And of course I get the answer
zero, and this is exactly what's
287
00:16:43 --> 00:16:47.93
happening in that,
in that, row times this wrong
288
00:16:47.93 --> 00:16:48
column.
289
00:16:48 --> 00:16:48
OK.
290
00:16:48 --> 00:16:54.47
That's the two by two picture,
and it's just the same here.
291
00:16:54.47 --> 00:16:59
That the reason I get a zero up
in there
292
00:16:59 --> 00:17:04
is, the reason I get a zero is
that when I multiply the first
293
00:17:04 --> 00:17:09
row of A and the last row of the
cofactor matrix,
294
00:17:09 --> 00:17:14
it's as if I'm taking the
determinant of this screwed up
295
00:17:14 --> 00:17:18
matrix that has first and last
rows identical.
296
00:17:18 --> 00:17:24
The book pins this down more
specific -- and more carefully
297
00:17:24 --> 00:17:27
than I can do in the lecture.
298
00:17:27 --> 00:17:30
I hope you're seeing the point.
299
00:17:30 --> 00:17:32
That this is an identity.
300
00:17:32 --> 00:17:36
That it's a beautiful identity
and it tells us what the inverse
301
00:17:36 --> 00:17:38
of the matrix is.
302
00:17:38 --> 00:17:42
So it gives us the inverse,
the formula for the inverse.
303
00:17:42 --> 00:17:42
OK.
304
00:17:42 --> 00:17:47
So that's the first goal of my
lecture, was to find this
305
00:17:47 --> 00:17:47
formula.
306
00:17:47 --> 00:17:48
It's done.
307
00:17:48 --> 00:17:48.76
OK.
308
00:17:48.76 --> 00:17:52
And of course I
could invert,
309
00:17:52 --> 00:17:57
now, I can, I sort of like I
can see what -- I can answer
310
00:17:57 --> 00:17:59
questions like this.
311
00:17:59 --> 00:18:04
Suppose I have a matrix,
and let me move the one one
312
00:18:04 --> 00:18:04
entry.
313
00:18:04 --> 00:18:07
What happens to the inverse?
314
00:18:07 --> 00:18:11
Just, just think about that
question.
315
00:18:11 --> 00:18:16
Suppose I have some
matrix, I just write down some
316
00:18:16 --> 00:18:20
nice, non-singular matrix that's
got an inverse,
317
00:18:20 --> 00:18:23
and then I move the one one
entry a little bit.
318
00:18:23 --> 00:18:25
I add one to it,
for example.
319
00:18:25 --> 00:18:28
What happens to the inverse
matrix?
320
00:18:28 --> 00:18:31
Well, this formula should tell
me.
321
00:18:31 --> 00:18:34
I have to look to see what
happens
322
00:18:34 --> 00:18:40
to the determinant and what
happens to all the cofactors.
323
00:18:40 --> 00:18:44
And, the picture,
it's all there.
324
00:18:44 --> 00:18:45
It's all there.
325
00:18:45 --> 00:18:51
We can really understand how
the inverse changes when the
326
00:18:51 --> 00:18:53
matrix changes.
327
00:18:53 --> 00:18:53.37
OK.
328
00:18:53.37 --> 00:18:59
Now my second application is to
-- let me put that over here --
329
00:18:59 --> 00:19:02
is to Ax=b.
330
00:19:02 --> 00:19:07
Well, the -- course,
the solution is A inverse b.
331
00:19:07 --> 00:19:11
But now I have a formula for A
inverse.
332
00:19:11 --> 00:19:16
A inverse is one over the
determinant times this C
333
00:19:16 --> 00:19:18.76
transpose times B.
334
00:19:18.76 --> 00:19:21
I now know what A inverse is.
335
00:19:21 --> 00:19:26
So now I just have to say,
what have I got here?
336
00:19:26 --> 00:19:32
Is there any way to,
to make this formula,
337
00:19:32 --> 00:19:35
this answer,
which is the one and only
338
00:19:35 --> 00:19:41
answer -- it's the very same
answer we got on the first day
339
00:19:41 --> 00:19:43
of the class by elimination.
340
00:19:43 --> 00:19:48
Now I'm -- now I've got a
formula for the answer.
341
00:19:48 --> 00:19:52
Can I play with it further to
see what's going on?
342
00:19:52 --> 00:19:57
And Cramer's,
this Cramer's Rule is
343
00:19:57 --> 00:20:02
exactly, that -- a way of
looking at this formula.
344
00:20:02 --> 00:20:02
OK.
345
00:20:02 --> 00:20:05.08
So this is a formula for x.
346
00:20:05.08 --> 00:20:06
Here's my formula.
347
00:20:06 --> 00:20:08
Well, of course.
348
00:20:08 --> 00:20:14
The first thing I see from the
formula is that the answer x
349
00:20:14 --> 00:20:17
always has that in the
determinant.
350
00:20:17 --> 00:20:19
I'm not surprised.
351
00:20:19 --> 00:20:24
There's a division by the
determinant.
352
00:20:24 --> 00:20:28.22
But then I have to say a little
more carefully what's going on
353
00:20:28.22 --> 00:20:28
up here.
354
00:20:28 --> 00:20:31
And let me tell you what
Cramer's Rule is.
355
00:20:31 --> 00:20:34
Let, let me take x1,
the first component.
356
00:20:34 --> 00:20:36
So this is the first component
of the answer.
357
00:20:36 --> 00:20:40
Then there'll be a second
component and a,
358
00:20:40 --> 00:20:42
all the other components.
359
00:20:42 --> 00:20:47
Can I take just the first
component of this formula?
360
00:20:47 --> 00:20:52
Well, I certainly have
determinant of A down under.
361
00:20:52 --> 00:20:57
And what the heck is the first
-- so what do I get in C
362
00:20:57 --> 00:20:58
transpose b?
363
00:20:58 --> 00:21:02
What's the first entry of C
transpose b?
364
00:21:02 --> 00:21:06
That's what I have to answer
myself.
365
00:21:06 --> 00:21:11.42
Well, what's the first entry of
C transpose b?
366
00:21:11.42 --> 00:21:11
OK.
367
00:21:11 --> 00:21:15
This B is -- let me tell you
what it is.
368
00:21:15 --> 00:21:21.07
Somehow I'm multiplying
cofactors by the entries of B,
369
00:21:21.07 --> 00:21:23
right, in this product.
370
00:21:23 --> 00:21:28
Cofactors from the matrix times
entries of b.
371
00:21:28 --> 00:21:33
So any time I'm multiplying
cofactors by numbers,
372
00:21:33 --> 00:21:37
I think, I'm getting the
determinant of something.
373
00:21:37 --> 00:21:40
And let me call that something
B1.
374
00:21:40 --> 00:21:44
So this is a matrix,
the matrix whose determinant is
375
00:21:44 --> 00:21:46
coming out of that.
376
00:21:46 --> 00:21:49
And we'll,
we'll see what it is.
377
00:21:49 --> 00:21:54
x2 will be the determinant of
some other matrix B2,
378
00:21:54 --> 00:21:57
also divided by determinant of
A.
379
00:21:57 --> 00:22:01
So now I just -- Cramer just
had a good idea.
380
00:22:01 --> 00:22:06
He realized what matrix it was,
what these B1 and B2 and B3 and
381
00:22:06 --> 00:22:08
so on matrices were.
382
00:22:08 --> 00:22:13
Let me write them on the board
underneath.
383
00:22:13 --> 00:22:13
OK.
384
00:22:13 --> 00:22:16
So what is this B1?
385
00:22:16 --> 00:22:24
This B1 is the matrix that has
b in its first column and
386
00:22:24 --> 00:22:28
otherwise the rest of it is A.
387
00:22:28 --> 00:22:37
So it otherwise it has the
rest, the, the n-1 columns of A.
388
00:22:37 --> 00:22:45
It's the matrix with -- it's
just the matrix A with column
389
00:22:45 --> 00:22:51
one replaced by the
right-hand side,
390
00:22:51 --> 00:22:54
by the right-hand side b.
391
00:22:54 --> 00:23:00
Because somehow when I take the
determinant of this guy,
392
00:23:00 --> 00:23:04
it's giving me this matrix
multiplication.
393
00:23:04 --> 00:23:07
Well, how could that be?
394
00:23:07 --> 00:23:13
How could -- so what's,
what's the determinant formula
395
00:23:13 --> 00:23:14.75
I'll use here?
396
00:23:14.75 --> 00:23:19
I'll use cofactors,
of course.
397
00:23:19 --> 00:23:23
And I might as well use
cofactors down column one.
398
00:23:23 --> 00:23:29
So when I use cofactors down
column one, I'm taking the first
399
00:23:29 --> 00:23:31
entry of b times what?
400
00:23:31 --> 00:23:33
Times the cofactor c11.
401
00:23:33 --> 00:23:35
Do you see that?
402
00:23:35 --> 00:23:40
When I, when I use cofactors
here, I take the first entry
403
00:23:40 --> 00:23:43
here,
B one let's call it,
404
00:23:43 --> 00:23:45
times the cofactor there.
405
00:23:45 --> 00:23:50.59
But what's the cofactor in --
my little hand-waving is meant
406
00:23:50.59 --> 00:23:54
to indicate that it's a matrix
of one size smaller,
407
00:23:54 --> 00:23:55
the cofactor.
408
00:23:55 --> 00:23:57
And it's exactly c11.
409
00:23:57 --> 00:23:59
Well, that's just what we
wanted.
410
00:23:59 --> 00:24:03
This first entry is c11
times b1.
411
00:24:03 --> 00:24:08.24
And then the next entry is
whatever, is c21 times b2,
412
00:24:08.24 --> 00:24:09
and so on.
413
00:24:09 --> 00:24:12.03
And sure enough,
if I look here,
414
00:24:12.03 --> 00:24:15
when I'm, when I do the
cofactor expansion,
415
00:24:15 --> 00:24:20
b2 is getting multiplied by the
right thing, and so on.
416
00:24:20 --> 00:24:23
So there's Cramer's Rule.
417
00:24:23 --> 00:24:27.47
And the book gives another kind
of
418
00:24:27.47 --> 00:24:34
cute proof without,
without building so much on,
419
00:24:34 --> 00:24:36
on cofactors.
420
00:24:36 --> 00:24:44
But here we've got cofactors,
so I thought I'd just give you
421
00:24:44 --> 00:24:46
this proof.
422
00:24:46 --> 00:24:52
So what is B -- in general,
what is Bj?
423
00:24:52 --> 00:24:57
This is the,
this is A with column j
424
00:24:57 --> 00:25:01
replaced by, by b.
425
00:25:01 --> 00:25:07
So that's -- the determinant of
that matrix that you divide by
426
00:25:07 --> 00:25:10
determinant of A to get xj.
427
00:25:10 --> 00:25:15
So x -- let me change this
general formula.
428
00:25:15 --> 00:25:19
xj, the j-th one,
is the determinant of Bj
429
00:25:19 --> 00:25:22
divided by the determinant of A.
430
00:25:22 --> 00:25:26.64
And now we've said what Bj is.
431
00:25:26.64 --> 00:25:29
Well, so Cramer found a rule.
432
00:25:29 --> 00:25:33
And we could ask him,
OK, great, good work,
433
00:25:33 --> 00:25:34
Cramer.
434
00:25:34 --> 00:25:38
But is your rule any good in
practice?
435
00:25:38 --> 00:25:42
So he says, well,
you couldn't ask about a rule
436
00:25:42 --> 00:25:47
in mine, right,
because it's just -- all you
437
00:25:47 --> 00:25:54
have to do is find the
determinant of A and these other
438
00:25:54 --> 00:25:57
determinants,
so I guess -- oh,
439
00:25:57 --> 00:26:02
he just says,
well, all you have to do is
440
00:26:02 --> 00:26:07
find n+1 determinants,
the, the n Bs and the A.
441
00:26:07 --> 00:26:12
And actually,
I remember reading -- there was
442
00:26:12 --> 00:26:20
a book, popular book that,
that kids interested in math
443
00:26:20 --> 00:26:24
read when I was a kid
interested in math called
444
00:26:24 --> 00:26:29
Mathematics for the Million or
something, by a guy named Bell.
445
00:26:29 --> 00:26:32
And it had a little page about
linear algebra.
446
00:26:32 --> 00:26:37
And it said,-- so it explained
elimination in a very
447
00:26:37 --> 00:26:38.51
complicated way.
448
00:26:38.51 --> 00:26:41
I certainly didn't understand
it.
449
00:26:41 --> 00:26:44
And, and it made it,
you know, it
450
00:26:44 --> 00:26:48
sort of said,
well, there is this formula for
451
00:26:48 --> 00:26:52
elimination, but look at this
great formula,
452
00:26:52 --> 00:26:53
Cramer's Rule.
453
00:26:53 --> 00:26:58
So it certainly said Cramer's
Rule was the way to go.
454
00:26:58 --> 00:27:02
But actually,
Cramer's Rule is a disastrous
455
00:27:02 --> 00:27:05
way to go, because to compute
these
456
00:27:05 --> 00:27:08
determinants,
it takes, like,
457
00:27:08 --> 00:27:10
approximately forever.
458
00:27:10 --> 00:27:15
So actually I now think of that
book title as being Mathematics
459
00:27:15 --> 00:27:19
for the Millionaire,
because you'd have to be able
460
00:27:19 --> 00:27:24
to pay for, a hopelessly long
calculation
461
00:27:24 --> 00:27:27
where elimination,
of course, produced the x-s,
462
00:27:27 --> 00:27:28
in an instant.
463
00:27:28 --> 00:27:33
But having a formula allows you
to, with, with letters,
464
00:27:33 --> 00:27:36
you know, allows you to do
algebra instead of,
465
00:27:36 --> 00:27:37
algorithms.
466
00:27:37 --> 00:27:42
So the, there's some value in
the Cramer's Rule formula for x
467
00:27:42 --> 00:27:47
and in the explicit formula
for, for A inverse.
468
00:27:47 --> 00:27:52.42
They're nice formulas,
but I just don't want you to
469
00:27:52.42 --> 00:27:53.39
use them.
470
00:27:53.39 --> 00:27:56
That'ss what it comes to.
471
00:27:56 --> 00:28:01
If you had to -- and Matlab
would never, never do it.
472
00:28:01 --> 00:28:06
I mean, it would use
elimination.
473
00:28:06 --> 00:28:06
OK.
474
00:28:06 --> 00:28:11
Now I'm ready for number three
in today's list of amazing
475
00:28:11 --> 00:28:16
connections coming through the
determinant.
476
00:28:16 --> 00:28:22
And that number three is the
fact that the determinant gives
477
00:28:22 --> 00:28:22
a volume.
478
00:28:22 --> 00:28:23
OK.
479
00:28:23 --> 00:28:29
So now -- so that's my final
topic for -- among these --
480
00:28:29 --> 00:28:33
this my number three
application, that the
481
00:28:33 --> 00:28:39
determinant is actually equals
the volume of something.
482
00:28:39 --> 00:28:44
Can I use this little space to
consider a special case,
483
00:28:44 --> 00:28:50
and then I'll use the far board
to think about the general rule.
484
00:28:50 --> 00:28:53
So what I going to prove?
485
00:28:53 --> 00:28:54
Or claim.
486
00:28:54 --> 00:29:00
I claim that the determinant of
the matrix is the volume of a
487
00:29:00 --> 00:29:01
box.
488
00:29:01 --> 00:29:03
OK, and you say,
which box?
489
00:29:03 --> 00:29:04
Fair enough.
490
00:29:04 --> 00:29:05
OK.
491
00:29:05 --> 00:29:06
So let's see.
492
00:29:06 --> 00:29:11
I'm in -- shall we say we're
in, say three by three?
493
00:29:11 --> 00:29:18
Shall we suppose -- let's,
let's say three by three.
494
00:29:18 --> 00:29:23
So, so we can really -- we're,
we're talking about boxes in
495
00:29:23 --> 00:29:27
three dimensions,
and three by three matrices.
496
00:29:27 --> 00:29:32
And so all I do -- you could
guess what the box is.
497
00:29:32 --> 00:29:35
Here is, here is,
three dimensions.
498
00:29:35 --> 00:29:35
OK.
499
00:29:35 --> 00:29:41
Now I take the first row of the
matrix, a11, a22,
500
00:29:41 --> 00:29:43
A -- sorry.
a11, a12, a13.
501
00:29:43 --> 00:29:44
That row is a vector.
502
00:29:44 --> 00:29:46
It goes to some point.
503
00:29:46 --> 00:29:50
That point will be -- and that
edge going to it,
504
00:29:50 --> 00:29:55
will be an edge of the box,
and that point will be a corner
505
00:29:55 --> 00:29:56
of the box.
506
00:29:56 --> 00:30:00
So here is zero zero zero,
of course.
507
00:30:00 --> 00:30:05
And here's the first row of the
matrix: a11, a12,
508
00:30:05 --> 00:30:06
a13.
509
00:30:06 --> 00:30:09
So that's one edge of the box.
510
00:30:09 --> 00:30:16
Another edge of the box is to
the second row of the matrix,
511
00:30:16 --> 00:30:17
row two.
512
00:30:17 --> 00:30:20.93
Can I just call it there row
two?
513
00:30:20.93 --> 00:30:28
And a third row of the box will
be to -- a third row --
514
00:30:28 --> 00:30:33
a third edge of the box will be
given by row three.
515
00:30:33 --> 00:30:35
So, so there's row three.
516
00:30:35 --> 00:30:40
That, the coordinates,
what are the coordinates of
517
00:30:40 --> 00:30:43
that corner of the box?
a31, a32, a33.
518
00:30:43 --> 00:30:48
So I've got that edge of the
box, that edge of the box,
519
00:30:48 --> 00:30:54
that edge of
the box, and that's all I need.
520
00:30:54 --> 00:30:58
Now I just finish out the box,
right?
521
00:30:58 --> 00:31:04
I just -- the proper word,
of course, is parallelepiped.
522
00:31:04 --> 00:31:08
But for obvious reasons,
I wrote box.
523
00:31:08 --> 00:31:09
OK.
524
00:31:09 --> 00:31:10
So, OK.
525
00:31:10 --> 00:31:16
So there's the,
there's the bottom of the box.
526
00:31:16 --> 00:31:20
There're the four edge sides of
the box.
527
00:31:20 --> 00:31:23
There's the top of the box.
528
00:31:23 --> 00:31:24
Cute, right?
529
00:31:24 --> 00:31:29
It's the box that has these
three edges and then it's
530
00:31:29 --> 00:31:33
completed to a,
to a, each, you know,
531
00:31:33 --> 00:31:37
each side is a,
is a parallelogram.
532
00:31:37 --> 00:31:43
And it's that box whose volume
is given by the determinant.
533
00:31:43 --> 00:31:49
That's -- now it's --
possible that the determinant
534
00:31:49 --> 00:31:50
is negative.
535
00:31:50 --> 00:31:55
So we have to just say what's
going on in that case.
536
00:31:55 --> 00:31:59
If the determinant is negative,
then the volume,
537
00:31:59 --> 00:32:03
we, we should take the absolute
value really.
538
00:32:03 --> 00:32:08
So the volume,
if we, if we think of volume as
539
00:32:08 --> 00:32:11
positive, we should take the
absolute value of the
540
00:32:11 --> 00:32:12
determinant.
541
00:32:12 --> 00:32:14
But the, the sign,
what does the sign of the
542
00:32:14 --> 00:32:17
determinant -- it always must
tell us something.
543
00:32:17 --> 00:32:20
And somehow it,
it will tell us whether these
544
00:32:20 --> 00:32:23
three is a -- whether it's a
right-handed box or a
545
00:32:23 --> 00:32:24
left-handed box.
546
00:32:24 --> 00:32:27
If we, if we reversed two of
the
547
00:32:27 --> 00:32:31
edges, we would go between a
right-handed box and a
548
00:32:31 --> 00:32:33
left-handed box.
549
00:32:33 --> 00:32:38
We wouldn't change the volume,
but we would change the,
550
00:32:38 --> 00:32:39
the cyclic, order.
551
00:32:39 --> 00:32:42
So I won't worry about that.
552
00:32:42 --> 00:32:45
And, so one special case is
what?
553
00:32:45 --> 00:32:47
A equal identity matrix.
554
00:32:47 --> 00:32:50.09
So let's take that special
case.
555
00:32:50.09 --> 00:32:52
A equal identity matrix.
556
00:32:52 --> 00:32:57
Is the
formula determinant of identity
557
00:32:57 --> 00:33:01
matrix, does that equal the
volume of the box?
558
00:33:01 --> 00:33:04
Well, what is the box?
559
00:33:04 --> 00:33:05
What's the box?
560
00:33:05 --> 00:33:11
If A is the identity matrix,
then these three rows are the
561
00:33:11 --> 00:33:17.25
three coordinate vectors,
and the box is -- it's a cube.
562
00:33:17.25 --> 00:33:20
It's the unit cube.
563
00:33:20 --> 00:33:24.4
So if, if A is the identity
matrix, of course our formula is
564
00:33:24.4 --> 00:33:24
right.
565
00:33:24 --> 00:33:28
Well, actually that proves
property one -- that the volume
566
00:33:28 --> 00:33:30
has property one.
567
00:33:30 --> 00:33:33
Actually, we could,
we could, we could get this
568
00:33:33 --> 00:33:37
thing if we -- if we can show
that the box volume has the same
569
00:33:37 --> 00:33:41
three properties that define the
determinant,
570
00:33:41 --> 00:33:43
then it must be the
determinant.
571
00:33:43 --> 00:33:47
So that's like the,
the, the elegant way to prove
572
00:33:47 --> 00:33:47.91
this.
573
00:33:47.91 --> 00:33:52
To prove this amazing fact that
the determinant equals the
574
00:33:52 --> 00:33:56
volume, first we'll check it for
the identity matrix.
575
00:33:56 --> 00:33:57
That's fine.
576
00:33:57 --> 00:34:00
The box is a cube and its
volume is one and the
577
00:34:00 --> 00:34:04
determinant is one and,
and one agrees with one.
578
00:34:04 --> 00:34:08
Now let me
take one -- let me go up one
579
00:34:08 --> 00:34:10
level to an orthogonal matrix.
580
00:34:10 --> 00:34:15
Because I'd like to take this
chance to bring in chapter --
581
00:34:15 --> 00:34:18.06
the, the previous chapter.
582
00:34:18.06 --> 00:34:21
Suppose I have an orthogonal
matrix.
583
00:34:21 --> 00:34:22
What did that mean?
584
00:34:22 --> 00:34:25.26
I always called those things Q.
585
00:34:25.26 --> 00:34:28
What was the point of --
suppose I
586
00:34:28 --> 00:34:35
have, suppose instead of the
identity matrix I'm now going to
587
00:34:35 --> 00:34:39
take A equal Q,
an orthogonal matrix.
588
00:34:39 --> 00:34:41
What was Q then?
589
00:34:41 --> 00:34:47
That was a matrix whose columns
were orthonormal,
590
00:34:47 --> 00:34:47.69
right?
591
00:34:47.69 --> 00:34:52
Those were its columns were
unit vectors,
592
00:34:52 --> 00:34:56
perpendicular unit vectors.
593
00:34:56 --> 00:34:59
So what kind of a box have we
got now?
594
00:34:59 --> 00:35:04
What kind of a box comes from
the rows or the columns,
595
00:35:04 --> 00:35:07
I don't mind,
because the determinant is the
596
00:35:07 --> 00:35:12
determinant of the transpose,
so I'm never worried about
597
00:35:12 --> 00:35:12
that.
598
00:35:12 --> 00:35:16.84
What kind of a box,
what shape box have we got if
599
00:35:16.84 --> 00:35:20
the matrix is an
orthogonal matrix?
600
00:35:20 --> 00:35:22
It's another cube.
601
00:35:22 --> 00:35:24
It's a cube again.
602
00:35:24 --> 00:35:28.41
How is it different from the
identity cube?
603
00:35:28.41 --> 00:35:30
It's just rotated.
604
00:35:30 --> 00:35:35
It's just the orthogonal matrix
Q doesn't have to be the
605
00:35:35 --> 00:35:36
identity matrix.
606
00:35:36 --> 00:35:41
It's just the unit cube but
turned in space.
607
00:35:41 --> 00:35:45
So sure enough,
it's the unit cube,
608
00:35:45 --> 00:35:47.41
and its volume is one.
609
00:35:47.41 --> 00:35:49
Now is the determinant one?
610
00:35:49 --> 00:35:52
What's the determinant of Q?
611
00:35:52 --> 00:35:57
We believe that the determinant
of Q better be one or minus one,
612
00:35:57 --> 00:36:01
so that our formula is --
checks out in that -- if we
613
00:36:01 --> 00:36:06
can't check it in these easy
cases where we got a cube,
614
00:36:06 --> 00:36:09
we're not going to get it in
the
615
00:36:09 --> 00:36:10
general case.
616
00:36:10 --> 00:36:15
So why is the determinant of Q
equal one or minus one?
617
00:36:15 --> 00:36:17
What do we know about Q?
618
00:36:17 --> 00:36:22
What's the one matrix statement
of the properties of Q?
619
00:36:22 --> 00:36:27
A matrix with orthonormal
columns has -- satisfies a
620
00:36:27 --> 00:36:29
certain equation.
621
00:36:29 --> 00:36:30
What, what is that?
622
00:36:30 --> 00:36:35
It's if we
have this orthogonal matrix,
623
00:36:35 --> 00:36:42
then the fact -- the way to say
what it, what its properties are
624
00:36:42 --> 00:36:43
is this.
625
00:36:43 --> 00:36:47
Q prime, u- u- Q transpose Q
equals I.
626
00:36:47 --> 00:36:48
Right?
627
00:36:48 --> 00:36:54
That's what -- those are the
matrices that get the name Q,
628
00:36:54 --> 00:36:59
the matrices that
Q transpose Q is I.
629
00:36:59 --> 00:37:00
OK.
630
00:37:00 --> 00:37:04
Now from that,
tell me why is the determinant
631
00:37:04 --> 00:37:06
one or minus one.
632
00:37:06 --> 00:37:11
How do I, out of this fact --
this may even be a homework
633
00:37:11 --> 00:37:12
problem.
634
00:37:12 --> 00:37:17
It's there in the,
in the list of exercises in the
635
00:37:17 --> 00:37:20
book, and let's just do it.
636
00:37:20 --> 00:37:24
How do I get,
how do I discover
637
00:37:24 --> 00:37:29
that the determinant of Q is
one or maybe minus one?
638
00:37:29 --> 00:37:33
I take determinants of both
sides, everybody says,
639
00:37:33 --> 00:37:37
so I won't -- I take
determinants of both sides.
640
00:37:37 --> 00:37:41
On the right-hand side -- so I,
when I take determinants of
641
00:37:41 --> 00:37:44
both sides, let me just do it.
642
00:37:44 --> 00:37:49
Take the determinant of --
take determinants.
643
00:37:49 --> 00:37:52
Determinant of the identity is
one.
644
00:37:52 --> 00:37:55
What's the determinant of that
product?
645
00:37:55 --> 00:37:58
Rule nine is paying off now.
646
00:37:58 --> 00:38:04
The determinant of a product is
the determinant of this guy --
647
00:38:04 --> 00:38:08
maybe I'll put it,
I'll use that symbol for
648
00:38:08 --> 00:38:09
determinant.
649
00:38:09 --> 00:38:13
It's the determinant of that
guy
650
00:38:13 --> 00:38:16
times the determinant of the
other guy.
651
00:38:16 --> 00:38:21
And then what's the determinant
of Q transpose?
652
00:38:21 --> 00:38:24
It's the same as the
determinant of Q.
653
00:38:24 --> 00:38:26
Rule ten pays off.
654
00:38:26 --> 00:38:29
So this is just this thing
squared.
655
00:38:29 --> 00:38:35
So that determinant squared is
one and sure enough it's one or
656
00:38:35 --> 00:38:36
minus one.
657
00:38:36 --> 00:38:36
Great.
658
00:38:36 --> 00:38:41
So in these special cases of
cubes,
659
00:38:41 --> 00:38:46
we really do have determinant
equals volume.
660
00:38:46 --> 00:38:50
Now can I just push that to
non-cubes.
661
00:38:50 --> 00:38:56
Let me push it first to
rectangles, rectangular boxes,
662
00:38:56 --> 00:39:03
where I'm just multiplying the
e- the edges are -- let me keep
663
00:39:03 --> 00:39:10
all the ninety degree angles,
because those are -- that,
664
00:39:10 --> 00:39:14
that makes my life easy.
665
00:39:14 --> 00:39:16
And just stretch the edges.
666
00:39:16 --> 00:39:21
Suppose I stretch that first
edge, suppose this first edge I
667
00:39:21 --> 00:39:22
double.
668
00:39:22 --> 00:39:26
Suppose I double that first
edge, keeping the other edges
669
00:39:26 --> 00:39:27
the same.
670
00:39:27 --> 00:39:29
What happens to the volume?
671
00:39:29 --> 00:39:31
It doubles, right?
672
00:39:31 --> 00:39:34
We know that the volume of a
cube doubles.
673
00:39:34 --> 00:39:38
In fact, because we know that
the
674
00:39:38 --> 00:39:42
new cube would sit right on top
-- I mean, the new,
675
00:39:42 --> 00:39:46
the added cube would sit right
on -- would fit -- probably a
676
00:39:46 --> 00:39:51
geometer would say congruent or
something -- would go right in,
677
00:39:51 --> 00:39:52
in the other.
678
00:39:52 --> 00:39:53
We'd have two.
679
00:39:53 --> 00:39:55
We have two identical cubes.
680
00:39:55 --> 00:39:57
Total volume is now two.
681
00:39:57 --> 00:39:58
OK.
682
00:39:58 --> 00:40:04
So I want -- if I double an
edge, the volume doubles.
683
00:40:04 --> 00:40:07
What happens to the
determinant?
684
00:40:07 --> 00:40:11.75
If I double,
the first row of a matrix,
685
00:40:11.75 --> 00:40:17
what ch- ch- what's the effect
on the determinant?
686
00:40:17 --> 00:40:19
It also doubles,
right?
687
00:40:19 --> 00:40:22
And that was rule number 3a.
688
00:40:22 --> 00:40:30
Remember rule 3a was that if I,
I could, if I had a factor in,
689
00:40:30 --> 00:40:34
in row one, T,
I could factor it out.
690
00:40:34 --> 00:40:38
So if, if I have a factor two
in that row one,
691
00:40:38 --> 00:40:42
I can factor it out of the
determinant.
692
00:40:42 --> 00:40:47
It agrees with the -- the
volume of the box has that
693
00:40:47 --> 00:40:48
factor two.
694
00:40:48 --> 00:40:53
So, so volume satisfies this
property 3a.
695
00:40:53 --> 00:40:59
And now I really close,
but I -- but to get to the very
696
00:40:59 --> 00:41:04
end of this proof,
I have to get away from right
697
00:41:04 --> 00:41:04
angles.
698
00:41:04 --> 00:41:09
I have to allow the possibility
of, other angles.
699
00:41:09 --> 00:41:15.71
And -- or what's saying the
same thing, I have to check that
700
00:41:15.71 --> 00:41:18
the volume also satisfies 3b.
701
00:41:18 --> 00:41:24
So can I --
This is end of proof that the
702
00:41:24 --> 00:41:29
-- so I'm -- determinant of A
equals volume of box,
703
00:41:29 --> 00:41:32
and where I right now?
704
00:41:32 --> 00:41:37.25
This volume has properties,
properties one,
705
00:41:37.25 --> 00:41:38
no problem.
706
00:41:38 --> 00:41:44
If the box is the cube,
everything is -- if the box is
707
00:41:44 --> 00:41:49
the unit cube,
its volume is one.
708
00:41:49 --> 00:41:53
Property two was if I reverse
two rows, but that doesn't
709
00:41:53 --> 00:41:55
change the box.
710
00:41:55 --> 00:41:58
And it doesn't change the
absolute value,
711
00:41:58 --> 00:41:59
so no problem there.
712
00:41:59 --> 00:42:03
Property 3a was if I mul- you
remember what 3a was?
713
00:42:03 --> 00:42:07.53
So property one was about the
identity matrix.
714
00:42:07.53 --> 00:42:12.34
Property two was about a plus
or minus sign that I don't care
715
00:42:12.34 --> 00:42:13
about.
716
00:42:13 --> 00:42:17
Property 3a was a factor T in a
row.
717
00:42:17 --> 00:42:22
But now I've got property three
B to deal with.
718
00:42:22 --> 00:42:24
What was property 3b?
719
00:42:24 --> 00:42:28
This is a great way to review
these, properties.
720
00:42:28 --> 00:42:33
So that 3b, the property 3b
said -- let's do,
721
00:42:33 --> 00:42:36
let's do two by two.
722
00:42:36 --> 00:42:43
So said that if I had a+a',
b+b', c, d that this equaled
723
00:42:43 --> 00:42:44.13
what?
724
00:42:44.13 --> 00:42:47
So this is property 3b.
725
00:42:47 --> 00:42:52
This is the linearity in row
one by itself.
726
00:42:52 --> 00:42:59
So c d is staying the same,
and I can split this into a b
727
00:42:59 --> 00:43:01
and a' b'.
728
00:43:01 --> 00:43:06
That's property 3b,
at least in the two by two
729
00:43:06 --> 00:43:09
case.
730
00:43:09 --> 00:43:14
And what I -- I wanted now to
show that the volume,
731
00:43:14 --> 00:43:19
which two, two by two,
that means area,
732
00:43:19 --> 00:43:22
has this, has this property.
733
00:43:22 --> 00:43:28
Let me just emphasize that we
have got -- we're getting --
734
00:43:28 --> 00:43:34
this is a formula,
then, for the area of a
735
00:43:34 --> 00:43:36
parallelogram.
736
00:43:36 --> 00:43:41
The area of this parallelogram
-- can I just draw it?
737
00:43:41 --> 00:43:45
OK, here's the,
here's the parallelogram.
738
00:43:45 --> 00:43:47
I have the row a b.
739
00:43:47 --> 00:43:49
That's the first row.
740
00:43:49 --> 00:43:52
That's the point a b.
741
00:43:52 --> 00:43:56
And I tack on c d.
c d, coming out of here.
742
00:43:56 --> 00:44:01
And I complete the
parallelogram.
743
00:44:01 --> 00:44:06
So this is -- well,
I better make it look right.
744
00:44:06 --> 00:44:12.71
It's really this one that has
coordinates c d and this has
745
00:44:12.71 --> 00:44:17
coordinates -- well,
whatever the sum is.
746
00:44:17 --> 00:44:21
And of course starting at zero
zero.
747
00:44:21 --> 00:44:24
So we all know,
this is a+c,
748
00:44:24 --> 00:44:25
b+d.
749
00:44:25 --> 00:44:31
Rather than -- I'm pausing on
that proof for a minute just to
750
00:44:31 --> 00:44:33
going back to our formula.
751
00:44:33 --> 00:44:38
Because I want you to see that
unlike Cramer's Rule,
752
00:44:38 --> 00:44:44.29
that I wasn't that impressed
by, I'm very impressed by this
753
00:44:44.29 --> 00:44:47
formula for the area of a
parallelogram.
754
00:44:47 --> 00:44:50
And what's our formula?
755
00:44:50 --> 00:44:53.46
What,
what's the area of that
756
00:44:53.46 --> 00:44:54
parallelogram?
757
00:44:54 --> 00:44:58
If I had asked you that last
year, you would have said OK,
758
00:44:58 --> 00:45:03
the area of a parallelogram is
the base times the height,
759
00:45:03 --> 00:45:03
right?
760
00:45:03 --> 00:45:06
So you would have figured out
what this base,
761
00:45:06 --> 00:45:09.24
the -- how long that base was.
762
00:45:09.24 --> 00:45:14
It's like the square root of A
squared plus b squared.
763
00:45:14 --> 00:45:21
And then you would have figured
out how much is this height,
764
00:45:21 --> 00:45:22
whatever it is.
765
00:45:22 --> 00:45:24
It's horrible.
766
00:45:24 --> 00:45:27
This, I mean,
we got square roots,
767
00:45:27 --> 00:45:33.7
and in that height there would
be other revolting stuff.
768
00:45:33.7 --> 00:45:40
But now what's the formula that
we now know for the area?
769
00:45:40 --> 00:45:46
It's the determinant of our
little matrix.
770
00:45:46 --> 00:45:48
It's just ad-bc.
771
00:45:48 --> 00:45:50
No square roots.
772
00:45:50 --> 00:45:57
Totally rememberable,
because it's exactly a formula
773
00:45:57 --> 00:46:04
that we've been studying the
whole, for three lectures.
774
00:46:04 --> 00:46:05
OK.
775
00:46:05 --> 00:46:11
That's, you know,
that's the most important point
776
00:46:11 --> 00:46:15
I'm making here.
777
00:46:15 --> 00:46:19.31
Is that if you know the
coordinates of a box,
778
00:46:19.31 --> 00:46:23
of the corners,
then you have a great formula
779
00:46:23 --> 00:46:25
for the volume,
area or volume,
780
00:46:25 --> 00:46:30
that doesn't involve any
lengths or any angles or any
781
00:46:30 --> 00:46:36
heights, but just involves the
coordinates that you've got.
782
00:46:36 --> 00:46:38
And similarly,
what's the area of this
783
00:46:38 --> 00:46:39
triangle?
784
00:46:39 --> 00:46:44
Suppose I chop that off and say
what about -- because you might
785
00:46:44 --> 00:46:47
often be interested in a
triangle instead of a
786
00:46:47 --> 00:46:48
parallelogram.
787
00:46:48 --> 00:46:51
What's the area of this
triangle?
788
00:46:51 --> 00:46:54
Now there again,
everybody would have said the
789
00:46:54 --> 00:46:59
area of a triangle is
half the base times the height.
790
00:46:59 --> 00:47:03
And in some cases,
if you know the base that a,
791
00:47:03 --> 00:47:06
that's -- and the height,
that's fine.
792
00:47:06 --> 00:47:09
But here, we,
what we know is the coordinates
793
00:47:09 --> 00:47:11
of the corners.
794
00:47:11 --> 00:47:12
We know the vertices.
795
00:47:12 --> 00:47:15
And so what's the area of that
triangle?
796
00:47:15 --> 00:47:19
If I know these,
if I know a b,
797
00:47:19 --> 00:47:24
c d, and zero zero,
what's the area?
798
00:47:24 --> 00:47:28
It's just half,
so it's just half of this.
799
00:47:28 --> 00:47:36
So this is, this is a- a b -- a
d - b c for the parallelogram
800
00:47:36 --> 00:47:41
and one half of that,
one half of ad-bc for the
801
00:47:41 --> 00:47:42
triangle.
802
00:47:42 --> 00:47:47
So I mean,
this is a totally trivial
803
00:47:47 --> 00:47:50
remark, to say,
well, divide by two.
804
00:47:50 --> 00:47:53
But it's just that you more
often see triangles,
805
00:47:53 --> 00:47:58
and you feel you know the
formula for the area but the
806
00:47:58 --> 00:48:01
good formula for the area is
this one.
807
00:48:01 --> 00:48:05.97
And I'm just going to -- I'm
just going to say one more thing
808
00:48:05.97 --> 00:48:09
about the area of a triangle.
809
00:48:09 --> 00:48:15
It's just because it's -- you
know, it's so great to have a
810
00:48:15 --> 00:48:18
good formula for something.
811
00:48:18 --> 00:48:22
What if our triangle did not
start at zero zero?
812
00:48:22 --> 00:48:28
What if our triangle,
what if we had this -- what if
813
00:48:28 --> 00:48:33
we had -- so I'm coming back to
triangles again.
814
00:48:33 --> 00:48:38
But let me, let me put this
triangle somewhere,
815
00:48:38 --> 00:48:44
it's -- I'm staying with
triangles, I'm just in two
816
00:48:44 --> 00:48:50
dimensions, but I'm going to
allow you to give me any three
817
00:48:50 --> 00:48:51
corners.
818
00:48:51 --> 00:48:57
And in -- those six numbers
must determine the area.
819
00:48:57 --> 00:48:59
And what's the formula?
820
00:48:59 --> 00:49:04
The area is going to be,
it's going to be,
821
00:49:04 --> 00:49:09
there'll be that
half of a parallelogram.
822
00:49:09 --> 00:49:13.68
I mean, basically this can't be
completely new,
823
00:49:13.68 --> 00:49:14
right?
824
00:49:14 --> 00:49:19
We've got the area when -- we,
we know the area when this is
825
00:49:19 --> 00:49:20
zero zero.
826
00:49:20 --> 00:49:25
Now we just want to lift our
sight slightly and get the area
827
00:49:25 --> 00:49:29
when all th- so let me write
down what it,
828
00:49:29 --> 00:49:32
what it comes out to be.
829
00:49:32 --> 00:49:36
It turns out that if you do
this, x1 y1 and a 1,
830
00:49:36 --> 00:49:39
x2 y2 and a 1,
x3 y3 and a 1,
831
00:49:39 --> 00:49:40.73
that that works.
832
00:49:40.73 --> 00:49:44
That the determinant symbol,
of course.
833
00:49:44 --> 00:49:49.04
It's just -- if I gave you that
determinant to find,
834
00:49:49.04 --> 00:49:52
you might subtract this row
from
835
00:49:52 --> 00:49:53
this.
836
00:49:53 --> 00:49:55
It would kill that one.
837
00:49:55 --> 00:49:59
Subtract this row from this,
it would kill that one.
838
00:49:59 --> 00:50:03.46
Then you'd have a simple
determinant to do with
839
00:50:03.46 --> 00:50:07
differences, and it would --
this little subtraction,
840
00:50:07 --> 00:50:12
what I did was equivalent to
moving the triangle to start at
841
00:50:12 --> 00:50:13
the origin.
842
00:50:13 --> 00:50:17
I did it fast,
because time is up.
843
00:50:17 --> 00:50:21
And I didn't complete that
proof of 3b.
844
00:50:21 --> 00:50:27
I'll leave -- the book has a
carefully drawn figure to show
845
00:50:27 --> 00:50:29
why that works.
846
00:50:29 --> 00:50:34
But I hope you saw the main
point is that for area and
847
00:50:34 --> 00:50:38
volume, determinant gives a
great formula.
848
00:50:38 --> 00:50:38
OK.
849
00:50:38 --> 00:50:42
And
next lectures are about
850
00:50:42 --> 00:50:47
eigenvalues, so we're really
into the big stuff.
851
00:50:47 --> 00:50:50
Thanks.