WEBVTT
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OK, this lecture is
like the beginning
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of the second half
of this is to prove.
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this course because up to now
we paid a lot of attention
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to rectangular matrices.
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Now, concentrating
on square matrices,
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so we're at two big topics.
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The determinant of
a square matrix,
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so this is the first
lecture in that new chapter
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on determinants, and the
reason, the big reason
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we need the determinants
is for the Eigen values.
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So this is really
determinants and Eigen values,
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the next big, big
chunk of 18.06.
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OK, so the determinant
is a number associated
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with every square matrix,
so every square matrix
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has this number associated with
called the, its determinant.
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I'll often write it as D E T A
or often also I'll write it as,
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A with vertical bars,
so that's going to mean
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the determinant of the matrix.
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That's going to mean this
one, like, magic number.
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Well, one number can't tell
you what the whole matrix was.
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But this one number, just
packs in as much information
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as possible into
a single number,
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and of course the one fact
that you've seen before
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and we have to see it
again is the matrix
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is invertible when the
determinant is not zero.
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The matrix is singular when
the determinant is zero.
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So the determinant will be
a test for invertibility,
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but the determinant's got
a lot more to it than that,
00:02:07.430 --> 00:02:08.889
so let me start.
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OK, now the question
is how to start.
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Do I give you a big formula
for the determinant,
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all in one gulp?
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I don't think so!
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That big formula has got
too much packed in it.
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I would rather start with three
properties of the determinant,
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three properties that it has.
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And let me tell
you property one.
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The determinant of
the identity is one.
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OK.
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I...
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I wish the other
two properties were
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as easy to tell you as that.
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But actually the second property
is pretty straightforward too,
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and then once we get the
third we will actually
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have the determinant.
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Those three properties define
the determinant and we can --
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then we can figure out, well,
what is the determinant?
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What's a formula for it?
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Now, the second property
is what happens if you
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exchange two rows of a matrix.
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What happens to the determinant?
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So, property two
is exchange rows,
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reverse the sign
of the determinant.
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A lot of plus and minus signs.
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I even wrote here,
"plus and minus signs,"
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because this is,
like, that's what
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you have to pay attention
to in the formulas
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and properties of determinants.
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So that -- you see what I
mean by a property here?
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I haven't yet told you
what the determinant is,
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but whatever it
is, if I exchange
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two rows to get a different
matrix that reverses
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the sign of the determinant.
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And so now, actually,
what matrices
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do we now know the
determinant of?
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From one and two, I now
know the determinant.
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Well, I certainly know the
determinant of the identity
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matrix and now I
know the determinant
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of every other matrix that
comes from row exchanges
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from the identities still.
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So what matrices have
I gotten at this point?
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The permutations, right.
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At this point I know
every permutation matrix,
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so now I'm saying the
determinant of a permutation
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matrix is one or minus one.
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One or minus one, depending
whether the number of exchanges
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was even or the number
of exchanges was odd.
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So this is the determinant
of a permutation.
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Now, P is back to
standing for permutation.
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OK.
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if I could carry on this
board, I could, like,
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do the two-by-two's.
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So, property one tells me
that this two-by-two matrix.
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Oh, I better write absolute --
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I mean, I'd better write
vertical bars, not brackets,
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for that determinant.
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Property one said, in
the two-by-two case,
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that this matrix
has determinant one.
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Property two tells me that
this matrix has determinant --
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what?
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Negative one.
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This is, like, two-by-twos.
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Now, I finally want to get --
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well, ultimately
I want to get to,
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the formula that we all know.
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Let me put that way over
here, that the determinant
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of a general
two-by-two is ad-bc.
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OK.
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I'm going to leave that up,
like, as the two by two case
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I'm down to the product of the
diagonal and if I transpose,
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that we already know,
so that every property,
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I can, like, check that it's
correct for two-by-twos.
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But the whole point
of these properties
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is that they're going to
give me a formula for n-by-n.
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That's the whole point.
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They're going to give
me this number that's
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a test for invertibility
and other great properties
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for any size matrix.
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OK, now you see I'm like,
slowing down because property
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three is the key property.
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Property three is the key
property and can I somehow
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describe it -- maybe I'll
separate it into 3A I said that
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if you do a row exchange,
the determinant and 3B.
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Property 3A says that if I
multiply one of the rows,
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say the first row,
by a number T --
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I'm going to erase that.
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That's, like, what I'm headed
for but I'm not there yet.
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It's the one we know
and you'll see that it's
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checked out by each property.
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OK, so this is for any matrix.
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For any matrix, if I
multiply one row by T
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and leave the other row
or other n-1 rows alone,
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what happens to the determinant?
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The factor T comes out.
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It's T times this determinant.
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That's not hard.
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I shouldn't have made a big
deal out of property 3A,
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and 3B is that, if
is, is if I keep --
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I'm always keeping this
second row the same,
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or that last n-1 rows
are all staying the same.
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I'm just working --
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I'm just looking inside
the first row and if I have
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an a+a' there and
a b+b' there --
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sorry, I didn't.
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Ahh.
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Why don't -- I'll use
an eraser, do it right.
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b+b' there.
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You see what I'm doing?
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This property and this property
are about linear combinations,
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of the first row only, leaving
the other rows unchanged.
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They'll copy along.
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Then, then I get the sum -- this
breaks up into the sum of this
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determinant and this one.
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I'm putting up formulas.
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Maybe I can try to
say it in words.
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The determinant is
a linear function.
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It behaves like a linear
function of first row
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if all the other
rows stay the same.
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I not saying that --
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let me emphasize.
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I not saying that the
determinant of A plus B
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is determinant of A
plus determinant of B.
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I not saying that.
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I'd better -- can I --
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how do I get it onto tape
that I'm not saying that?
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You see, this would allow
all the rows -- you know,
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A to have a bunch of rows,
B to have a bunch of rows.
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That's not the
linearity I'm after.
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I'm only after
linearity in each row.
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Linear for each row.
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Well, you may say I only
talked about the first row,
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but I claim it's also
linear in the second row,
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if I had this -- but not,
I can't, I can't have
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a combination in both
first and second.
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If I had a combination
in the second row,
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then I could use rule two to
put it up in the first row,
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use my property and then use
rule two again to put it back,
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so each row is OK, not
only the first row,
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but each row separately.
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OK, those are the
three properties,
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and from those
properties, so that's
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properties one, two, three.
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From those, I want to get all --
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I'm going to learn a lot
more about the determinant.
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Let me take an example.
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What would I like to learn?
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I would like to learn that --
so here's our property four.
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Let me use the same
numbering as here.
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Property four is if two rows are
equal, the determinant is zero.
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OK, so property four.
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Two equal rows lead to
determinant equals zero.
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Right.
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Now, of course I can -- in the
two-by-two case I can check,
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sure, the determinant
of ab ab comes out zero.
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But I want to see why
it's true for n-by-n.
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Suppose row one equals row three
for a seven-by-seven matrix.
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So two rows are the
same in a big matrix.
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And all I have to work
with is these properties.
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The exchange property,
which flips the sign,
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and the linearity property which
works in each row separately.
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OK, can you see the reason?
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How do I get this one out of
properties one, two, three?
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Because -- that's all
I have to work with.
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Everything has to come from
properties one, two, three.
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OK, so suppose I have a
matrix, and two rows are even.
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How do I see that
its determinant
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has to be zero from
these properties?
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I do an exchange.
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Property two is just
set up for this.
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Use property two.
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Use exchange -- exchange rows.
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Exchange those rows, and
I get the same matrix.
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Of course, because
those rows were equal.
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So the determinant
didn't change.
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But on the other
hand, property two
00:13:52.360 --> 00:13:56.170
says that the sign did change.
00:13:56.170 --> 00:13:59.350
So the -- so I, I have
a determinant whose sign
00:13:59.350 --> 00:14:03.390
doesn't change and does change,
and the only possibility then
00:14:03.390 --> 00:14:06.520
is that the determinant is zero.
00:14:06.520 --> 00:14:08.780
You see the reasoning there?
00:14:08.780 --> 00:14:09.550
Straightforward.
00:14:09.550 --> 00:14:15.250
Property two just told us, hey,
if we've got two equal rows we.
00:14:15.250 --> 00:14:19.210
we've got a zero determinant.
00:14:19.210 --> 00:14:22.140
And of course in our minds,
that matches the fact
00:14:22.140 --> 00:14:26.550
that if I have two equal rows
the matrix isn't invertible.
00:14:26.550 --> 00:14:29.430
If I have two equal rows,
I know that the rank
00:14:29.430 --> 00:14:31.190
is less changes sign. than n.
00:14:31.190 --> 00:14:34.540
OK, ready for property five.
00:14:34.540 --> 00:14:38.900
Now, property five
you'll recognize as P.
00:14:38.900 --> 00:14:45.710
It says that the elimination
step that I'm always doing,
00:14:45.710 --> 00:14:51.850
or U and U transposed, when
they're triangular,4 subtract
00:14:51.850 --> 00:15:00.560
a multiple, subtract some
multiple l times row one from
00:15:00.560 --> 00:15:04.800
another row, row k, let's say.
00:15:07.950 --> 00:15:10.760
You remember why I did that.
00:15:10.760 --> 00:15:14.620
In elimination I'm always
choosing this multiplier so as
00:15:14.620 --> 00:15:17.190
to produce zero
in that position.
00:15:17.190 --> 00:15:21.050
What I -- way, way
back in property two,4
00:15:21.050 --> 00:15:24.230
Or row I from row k,
maybe I should just
00:15:24.230 --> 00:15:28.250
make very clear that there's
nothing special about row one
00:15:28.250 --> 00:15:30.530
here.
00:15:30.530 --> 00:15:34.400
OK, so that, you can see
why I want that who cares?
00:15:34.400 --> 00:15:37.350
one, because that
will allow me to start
00:15:37.350 --> 00:15:40.890
with this full matrix whose
determinant I don't know,
00:15:40.890 --> 00:15:45.590
and I can do elimination
and clean it out.
00:15:45.590 --> 00:15:47.830
I can get zeroes
below the diagonal
00:15:47.830 --> 00:15:50.970
by these elimination
steps and the point
00:15:50.970 --> 00:15:58.420
is that the determinant, the
determinant doesn't change.
00:16:10.210 --> 00:16:12.450
So all those steps
of elimination
00:16:12.450 --> 00:16:15.070
are OK not changing
the determinant.
00:16:15.070 --> 00:16:18.580
In our language in the early
chapter the determinant of A is
00:16:18.580 --> 00:16:20.900
So if I do seven row
exchanges, the determinant
00:16:20.900 --> 00:16:23.989
changes sign, going to be the
same as the determinant of U,
00:16:23.989 --> 00:16:25.030
the upper triangular one.
00:16:25.030 --> 00:16:27.090
It just has the pivots
on the diagonal.
00:16:27.090 --> 00:16:29.100
That's why we'll
want this property.
00:16:29.100 --> 00:16:31.250
OK, do you see where that
property's coming from?
00:16:34.180 --> 00:16:37.110
Let me do the two-by-two case.
00:16:37.110 --> 00:16:39.490
Let me do the
two-by-two case here.
00:16:39.490 --> 00:16:44.340
So, I'll keep property
five going along.
00:16:44.340 --> 00:16:45.300
So what I doing?
00:16:45.300 --> 00:16:46.890
I'm going to keep --
00:16:46.890 --> 00:16:52.910
I'm going to have
ab cd, but I'm going
00:16:52.910 --> 00:16:57.460
to subtract l times the first
row from the second row.
00:16:57.460 --> 00:17:03.182
And that's the
determinant and of
00:17:03.182 --> 00:17:03.890
OK, that's not --
00:17:03.890 --> 00:17:06.260
I didn't put in every
comma and, course
00:17:06.260 --> 00:17:10.900
I can multiply that out and
figure out, sure enough, ad-bc
00:17:10.900 --> 00:17:17.849
is there and this minus
ALB plus ALB cancels out,
00:17:17.849 --> 00:17:19.569
but I just cheated,
00:17:19.569 --> 00:17:20.190
right?
00:17:20.190 --> 00:17:21.760
I've got to use the properties.
00:17:21.760 --> 00:17:22.510
So what property?
00:17:22.510 --> 00:17:24.599
Well, of course,
this is a com --
00:17:24.599 --> 00:17:28.420
I'm keeping the first row
the same and the second row
00:17:28.420 --> 00:17:31.690
has a c and a d,
and then there's
00:17:31.690 --> 00:17:36.190
the determinant of the A
and the B, and the minus LA,
00:17:36.190 --> 00:17:37.240
and the minus LB.
00:17:41.840 --> 00:17:44.150
So what property was that?
00:17:44.150 --> 00:17:46.420
3B.
00:17:46.420 --> 00:17:49.340
I kept one row
the same and I had
00:17:49.340 --> 00:17:52.690
a combination in the
second, in the other row,
00:17:52.690 --> 00:17:56.520
and I just separated it out.
00:17:56.520 --> 00:17:59.300
OK, so that's property 3.
00:17:59.300 --> 00:18:03.350
That's by number
3, 3B if you like.
00:18:03.350 --> 00:18:04.980
OK, now use 3A.
00:18:04.980 --> 00:18:10.480
How do you use 3A, which
says I can factor out an l,
00:18:10.480 --> 00:18:13.110
I can factor out a minus l here.
00:18:13.110 --> 00:18:17.140
I can factor a minus l out
from this row, no problem.
00:18:17.140 --> 00:18:19.000
That was 3A.
00:18:19.000 --> 00:18:25.390
So now I've used property three
and now I'm ready for the kill.
00:18:25.390 --> 00:18:32.070
Property four says that
this determinant is zero,
00:18:32.070 --> 00:18:34.810
has two equal rows.
00:18:34.810 --> 00:18:37.100
You see how that
would always work?
00:18:37.100 --> 00:18:40.280
I subtract a multiple of
one row from another one.
00:18:40.280 --> 00:18:46.860
It gives me a combination in
row k of the old row and l times
00:18:46.860 --> 00:18:51.310
this copy of the higher
row, and then if --
00:18:51.310 --> 00:18:53.550
since I have two equal
rows, that's zero,
00:18:53.550 --> 00:18:56.990
so the determinant after
elimination is the same
00:18:56.990 --> 00:18:58.280
as before.
00:18:58.280 --> 00:18:59.430
Good.
00:18:59.430 --> 00:19:00.310
OK.
00:19:00.310 --> 00:19:04.170
Now, let's see -- if
I rescue my glasses,
00:19:04.170 --> 00:19:07.140
I can see what's property six.
00:19:07.140 --> 00:19:11.700
Oh, six is easy,
plenty of space.
00:19:11.700 --> 00:19:22.450
Row of zeroes leads to
determinant of A equals zero.
00:19:26.840 --> 00:19:28.380
A complete row of zeroes.
00:19:28.380 --> 00:19:32.550
So I'm again, this
is like, something
00:19:32.550 --> 00:19:34.880
I'll use in the singular case.
00:19:34.880 --> 00:19:39.380
Actually, you can look ahead
to why I need these properties.
00:19:39.380 --> 00:19:42.220
So I'm going to use property
five, the elimination,
00:19:42.220 --> 00:19:45.980
use this stuff to say
that this determinant is
00:19:45.980 --> 00:19:49.600
the same as that determinant
and I'll produce a zero there.
00:19:49.600 --> 00:19:51.910
But what if I also
produce a zero there?
00:19:51.910 --> 00:19:54.620
What if elimination
gives a row of zeroes?
00:19:54.620 --> 00:19:59.120
That, that used to be my
test for, mmm, singular,
00:19:59.120 --> 00:20:03.220
not invertible, rank
two -- rank less than N,
00:20:03.220 --> 00:20:07.120
and now I'm seeing it's
also gives determinant zero.
00:20:07.120 --> 00:20:12.600
How do I get that one from
the previous properties?
00:20:12.600 --> 00:20:15.006
'Cause I -- this
is not a new law,
00:20:15.006 --> 00:20:16.630
this has got to come
from the old ones.
00:20:16.630 --> 00:20:20.635
So what shall I do?
00:20:23.210 --> 00:20:24.690
Yeah, use -- that's brilliant.
00:20:24.690 --> 00:20:26.430
If you use 3A with
T equals zero.
00:20:26.430 --> 00:20:27.350
Right.
00:20:27.350 --> 00:20:32.760
So I have this zero
zero cd, and I'm
00:20:32.760 --> 00:20:35.850
trying to show that that
determinant is zero. triangular
00:20:35.850 --> 00:20:37.560
matrices, l and l transposed,
00:20:37.560 --> 00:20:41.000
OK, so the zero is
the same is -- five,
00:20:41.000 --> 00:20:45.900
can I take T equals five,
just to, like, pin it down?
00:20:45.900 --> 00:20:48.530
I'll multiply this row by five.
00:20:48.530 --> 00:20:52.780
Five, well then, five
of this should -- if I,
00:20:52.780 --> 00:21:01.300
if there's a factor five
in that, you see what --
00:21:01.300 --> 00:21:05.500
so this is property 3A,
with taking T as five.
00:21:05.500 --> 00:21:08.660
If I multiply a row by
five, out comes a five.
00:21:08.660 --> 00:21:14.450
So why I doing this?
00:21:14.450 --> 00:21:19.320
Oh, because that's
still zero zero, right?
00:21:19.320 --> 00:21:21.000
So that's this
determinant equals
00:21:21.000 --> 00:21:28.780
five times this determinant, and
the determinant has to be zero.
00:21:28.780 --> 00:21:34.100
I think I didn't do
that the very best way.
00:21:34.100 --> 00:21:36.670
You were, yeah, you
had the idea better.
00:21:36.670 --> 00:21:40.620
Multiply, yeah,
take T equals zero.
00:21:40.620 --> 00:21:44.170
Was that your idea?
00:21:44.170 --> 00:21:46.840
Take T equals zero in rule 3B.
00:21:46.840 --> 00:21:51.990
If T is zero in rule 3B, and I
bring the camera back to rule
00:21:51.990 --> 00:21:52.720
3B --
00:21:52.720 --> 00:21:55.260
sorry.
00:21:55.260 --> 00:22:01.300
If T is zero, then I
have a zero zero there
00:22:01.300 --> 00:22:03.330
and the determinant is zero.
00:22:03.330 --> 00:22:08.710
OK, one way or another, a row of
zeroes means zero determinant.
00:22:08.710 --> 00:22:14.930
OK, now I have to get serious.
00:22:14.930 --> 00:22:20.500
The next properties are the
ones that we're building up to.
00:22:20.500 --> 00:22:23.750
OK, so I can do elimination.
00:22:23.750 --> 00:22:26.540
I can reduce to a
triangular matrix
00:22:26.540 --> 00:22:30.120
and now what's the determinant
of that triangular matrix?
00:22:30.120 --> 00:22:34.100
OK, so they had to wait
until the last minute.
00:22:34.100 --> 00:22:37.280
Suppose, suppose I --
all right, rule seven.
00:22:37.280 --> 00:22:42.430
So suppose my matrix
is now triangular.
00:22:42.430 --> 00:22:44.660
So it's got --
00:22:44.660 --> 00:22:48.870
so I even give these the names
of the pivots, d1, d2, to dn,
00:22:48.870 --> 00:22:54.710
and stuff is up here, I
don't know what that is,
00:22:54.710 --> 00:22:57.970
but what I do know is
this is all zeroes.
00:22:57.970 --> 00:23:04.750
That's all zeroes, and I would
like to know the determinant,
00:23:04.750 --> 00:23:08.370
because elimination
will get me to this.
00:23:08.370 --> 00:23:11.610
So once I'm here, what's
the determinant then?
00:23:11.610 --> 00:23:16.560
Let me use an eraser to get
those, get that vertical bar
00:23:16.560 --> 00:23:22.350
again, so that I'm taking the
determinant of U so that, so,
00:23:22.350 --> 00:23:28.510
what is the determinant of
an upper triangular matrix?
00:23:28.510 --> 00:23:33.215
Do you know the answer?
00:23:36.090 --> 00:23:40.220
It's just the product
of the d's. for it.
00:23:40.220 --> 00:23:44.740
The -- these things that I
don't even put in letters
00:23:44.740 --> 00:23:53.020
for, because they don't matter,
the determinant is d1 times d2
00:23:53.020 --> 00:23:54.197
times dn.
00:23:57.120 --> 00:24:02.100
If I have a triangular
matrix, then the diagonal
00:24:02.100 --> 00:24:04.770
is all I have to work with.
00:24:04.770 --> 00:24:06.550
And that's, that's
telling us then.
00:24:06.550 --> 00:24:15.120
We now have the way that
MATLAB, any reasonable software,
00:24:15.120 --> 00:24:17.250
would compute a determinant.
00:24:17.250 --> 00:24:20.990
If I have a matrix
of size a hundred,
00:24:20.990 --> 00:24:24.817
the way I would actually
compute its determinant would be
00:24:24.817 --> 00:24:27.400
elimination, make it triangular,
multiply the pivots together,
00:24:27.400 --> 00:24:29.940
but it -- would
it be possible t-
00:24:29.940 --> 00:24:33.110
to produce the same matrix
the product of the pivots,
00:24:33.110 --> 00:24:34.080
the product of pivots.
00:24:34.080 --> 00:24:39.080
Now, there's always in
determinants a plus or minus
00:24:39.080 --> 00:24:44.630
and cross every T in that
proof, but that's really
00:24:44.630 --> 00:24:46.850
the sign to remember.
00:24:46.850 --> 00:24:51.840
Where -- where does that
come into this rule?
00:24:51.840 --> 00:24:54.500
Could it be, could
the determinant
00:24:54.500 --> 00:24:58.210
be minus the product
of the pivots?
00:24:58.210 --> 00:25:00.740
The determinant
is d1, d2, to dn.
00:25:00.740 --> 00:25:02.380
No doubt about that.
00:25:02.380 --> 00:25:05.370
But to get to this
triangular form,
00:25:05.370 --> 00:25:12.660
we may have had to do a row
exchange, so, so this --
00:25:12.660 --> 00:25:15.890
it's the product of the pivots
if there were no row exchanges.
00:25:15.890 --> 00:25:19.370
If, if SLU code,
the simple LU code,
00:25:19.370 --> 00:25:21.770
the square LU went
right through.
00:25:21.770 --> 00:25:24.070
If we had to do
some row exchanges,
00:25:24.070 --> 00:25:26.840
then we've got to
watch plus or minus.
00:25:26.840 --> 00:25:31.680
OK, but though -- this
law is simply that.
00:25:31.680 --> 00:25:33.180
OK, how do I prove that?
00:25:37.420 --> 00:25:42.730
Let's see, let me suppose
that d's are not zeroes.
00:25:42.730 --> 00:25:44.620
The pivots are not zeroes.
00:25:44.620 --> 00:25:50.550
And tell me, how do I show
that none of this upper stuff
00:25:50.550 --> 00:25:53.840
makes any difference?
00:25:53.840 --> 00:25:56.550
How do I get zeroes there?
00:25:56.550 --> 00:25:58.630
By elimination, right?
00:25:58.630 --> 00:26:01.940
I just multiply this
row by the right number,
00:26:01.940 --> 00:26:05.950
subtract from that
row, kills that.
00:26:05.950 --> 00:26:09.030
I multiply this row by the
right number, kills that,
00:26:09.030 --> 00:26:11.010
by the right number, kills that.
00:26:11.010 --> 00:26:15.850
Now, you kill every one of these
off-diagonal terms, no problem
00:26:15.850 --> 00:26:17.450
and I'm just left
with the diagonal.
00:26:20.690 --> 00:26:22.940
Well, strictly
speaking, I still have
00:26:22.940 --> 00:26:26.680
to figure out why is,
for a diagonal matrix
00:26:26.680 --> 00:26:28.745
now, why is that the
right determinant?
00:26:31.540 --> 00:26:37.260
I mean, we sure
hope it is, but why?
00:26:37.260 --> 00:26:41.270
I have to go back to
properties one, two, three.
00:26:41.270 --> 00:26:46.710
Why is -- now that the
matrix is suddenly diagonal,
00:26:46.710 --> 00:26:48.950
how do I know that the
determinant is just
00:26:48.950 --> 00:26:51.070
a product of That's
my proof, really,
00:26:51.070 --> 00:26:53.200
that once I've got
those diagonal entries?
00:26:53.200 --> 00:26:55.030
Well, what I going to use?
00:26:55.030 --> 00:26:57.760
I'm going to use property
3A, is that right?
00:26:57.760 --> 00:27:01.250
I'll factor this,
I'll factor this,
00:27:01.250 --> 00:27:05.380
I'll factor that d1 out
and have one and have
00:27:05.380 --> 00:27:07.280
the first row will be that.
00:27:07.280 --> 00:27:09.710
And then I'll factor out
the d2, shall I shall
00:27:09.710 --> 00:27:13.160
I put the d2 here,
and the second row
00:27:13.160 --> 00:27:16.110
will look like that, and so on.
00:27:16.110 --> 00:27:20.150
So I've factored out all the
d's and what I left with?
00:27:20.150 --> 00:27:21.370
The identity.
00:27:21.370 --> 00:27:24.970
And what rule do I
finally get to use?
00:27:24.970 --> 00:27:26.030
Rule one.
00:27:26.030 --> 00:27:29.900
Finally, this is the point
where rule one finally chips
00:27:29.900 --> 00:27:33.480
in and says that this
determinant is one,
00:27:33.480 --> 00:27:35.520
so it's the product of the d's.
00:27:35.520 --> 00:27:40.950
So this was rules five,
to do elimination,
00:27:40.950 --> 00:27:48.680
3A to factor out the D's, and,
and our best friend, rule one.
00:27:48.680 --> 00:27:52.240
And possibly rule
two, the exchanges
00:27:52.240 --> 00:27:53.710
may have been needed also.
00:27:53.710 --> 00:27:54.210
OK.
00:27:56.940 --> 00:28:01.350
Now I guess I have to consider
also the case if some d is
00:28:01.350 --> 00:28:06.680
zero, because I was assuming I
could use the d's to clean out
00:28:06.680 --> 00:28:08.490
and make a diagonal,
but what if --
00:28:08.490 --> 00:28:13.390
what if one of those
diagonal entries is zero?
00:28:13.390 --> 00:28:16.380
Well, then with
elimination we know
00:28:16.380 --> 00:28:21.440
that we can get a row
of zeroes, and for a row
00:28:21.440 --> 00:28:25.200
of zeroes I'm using rule
six, the determinant is zero,
00:28:25.200 --> 00:28:26.020
and that's right.
00:28:26.020 --> 00:28:28.690
So I can check
the singular case.
00:28:28.690 --> 00:28:36.250
In fact, I can now get to the
key point that determinant of A
00:28:36.250 --> 00:28:44.920
is zero, exactly when,
exactly when A is singular.
00:28:48.790 --> 00:28:52.880
And otherwise is not singular,
so that the determinant
00:28:52.880 --> 00:28:58.800
is a fair test for invertibility
or non-invertibility.
00:28:58.800 --> 00:29:03.610
So, I must be close to that
because I can take any matrix
00:29:03.610 --> 00:29:05.290
and get there.
00:29:05.290 --> 00:29:06.870
Do I -- did I have
anything to say?
00:29:09.570 --> 00:29:12.970
The, the proofs, it starts
by saying by elimination
00:29:12.970 --> 00:29:14.680
go from A to U.
00:29:14.680 --> 00:29:15.290
Oh, yeah.
00:29:15.290 --> 00:29:17.940
Actually looks to
me like I don't --
00:29:17.940 --> 00:29:22.450
haven't said anything brand-new
here, that, that really,
00:29:22.450 --> 00:29:28.950
I've got this, because
let's just remember the
00:29:28.950 --> 00:29:37.980
By elimination, I can go from
the original A to reason.
00:29:37.980 --> 00:29:43.440
Well, OK, now suppose the
matrix is U. singular.
00:29:43.440 --> 00:29:46.630
If the matrix is
singular, what happens?
00:29:46.630 --> 00:29:50.480
Then by elimination
I get a row of zeroes
00:29:50.480 --> 00:29:55.240
and therefore the
determinant is zero.
00:29:55.240 --> 00:29:59.170
And if the matrix is not
singular, I don't get zero,
00:29:59.170 --> 00:30:02.220
so maybe -- do you want me to
put this, like, in two parts?
00:30:02.220 --> 00:30:10.100
The determinant of A is not
zero when A is invertible.
00:30:15.480 --> 00:30:18.750
Because I've already --
00:30:18.750 --> 00:30:23.240
in chapter two we figured out
when is the matrix invertible.
00:30:23.240 --> 00:30:27.770
It's invertible when elimination
produces a full set of pivots
00:30:27.770 --> 00:30:31.420
and now, and we now, we know
the determinant is the product
00:30:31.420 --> 00:30:34.160
of those non-zero numbers.
00:30:34.160 --> 00:30:36.410
So those are the two cases.
00:30:36.410 --> 00:30:39.180
If it's singular, I
go to a row of zeroes.
00:30:43.370 --> 00:30:49.680
If it's invertible, I go to
U and then to the diagonal D,
00:30:49.680 --> 00:30:57.150
and then which -- and
then to d1, d2, up to dn.
00:30:57.150 --> 00:31:00.050
As the formula -- so
we have a formula now.
00:31:02.700 --> 00:31:05.110
We have a formula
for the determinant
00:31:05.110 --> 00:31:08.520
and it's actually a
very much more practical
00:31:08.520 --> 00:31:12.100
formula than the but they didn't
matter anyway. ad-bc formula.
00:31:12.100 --> 00:31:18.050
Is it correct, maybe you should
just -- let's just check that.
00:31:18.050 --> 00:31:18.970
Two-by-two.
00:31:18.970 --> 00:31:23.850
What are the pivots of
a two-by-two matrix?
00:31:23.850 --> 00:31:28.270
What does elimination do
with a two-by-two matrix?
00:31:28.270 --> 00:31:30.370
It -- there's the
first pivot, fine.
00:31:30.370 --> 00:31:33.880
What's the second pivot?
00:31:33.880 --> 00:31:38.970
We subtract, so I'm putting it
in this upper triangular form.
00:31:38.970 --> 00:31:44.270
What do I -- my multiplier
is c over a, right?
00:31:44.270 --> 00:31:46.790
I multiply that row
by c over a and I
00:31:46.790 --> 00:31:50.590
subtract to get that
zero, and here I
00:31:50.590 --> 00:31:54.470
have d minus c over a times b.
00:31:58.500 --> 00:32:01.870
That's the elimination
on a two-by-two.
00:32:01.870 --> 00:32:07.440
So I've finally discovered that
the determinant of this matrix
00:32:07.440 --> 00:32:07.940
--
00:32:07.940 --> 00:32:12.240
I've got it from the properties,
not by knowing the answer
00:32:12.240 --> 00:32:18.830
from last year, that the
determinant of this two-by-two
00:32:18.830 --> 00:32:22.730
is the product of A
times that, and of course
00:32:22.730 --> 00:32:26.180
when I multiply A by that,
the product of that and that
00:32:26.180 --> 00:32:30.485
is ad minus, the a is canceled,
00:32:30.485 --> 00:32:30.985
bc.
00:32:34.270 --> 00:32:36.320
So that's great,
provided a isn't zero.
00:32:36.320 --> 00:32:38.960
because all math professors
watching this will be waiting
00:32:38.960 --> 00:32:42.190
If a was zero, that step
wasn't allowed, with seven row
00:32:42.190 --> 00:32:45.130
exchanges and with ten row
exchanges? zero wasn't a pivot.
00:32:45.130 --> 00:32:46.600
So that's what I --
00:32:46.600 --> 00:32:48.850
I've covered all the bases.
00:32:48.850 --> 00:32:53.090
I have to -- if a is zero,
then I have to do the exchange,
00:32:53.090 --> 00:32:56.570
and if the exchange doesn't
work, it's because a is proof.
00:32:56.570 --> 00:32:57.730
singular.
00:32:57.730 --> 00:33:03.130
OK, those are --
00:33:03.130 --> 00:33:06.520
those are the direct
properties of the determinant.
00:33:06.520 --> 00:33:11.120
And now, finally, I've got
two more, nine and ten.
00:33:11.120 --> 00:33:13.920
And that's --
00:33:13.920 --> 00:33:15.030
I think you can...
00:33:15.030 --> 00:33:25.210
Like, the ones
we've got here are
00:33:25.210 --> 00:33:28.870
totally connected with
our elimination process
00:33:28.870 --> 00:33:35.340
and whether pivots are
available and whether we
00:33:35.340 --> 00:33:36.820
get a row of zeroes.
00:33:36.820 --> 00:33:40.360
I think all that you
can swallow in one shot.
00:33:40.360 --> 00:33:43.730
Let me tell you
properties nine and ten.
00:33:46.990 --> 00:33:50.000
They're quick to write down.
00:33:50.000 --> 00:33:53.940
They're very, very useful.
00:33:53.940 --> 00:33:56.260
So I'll just write
them down and use them.
00:33:56.260 --> 00:34:01.410
Property nine says that the
determinant of a product --
00:34:01.410 --> 00:34:05.610
if I That's the, like,
concrete proof that,
00:34:05.610 --> 00:34:07.180
multiply two matrices.
00:34:07.180 --> 00:34:11.909
So if I multiply two
matrices, A and B,
00:34:11.909 --> 00:34:14.050
that the determinant
of the product
00:34:14.050 --> 00:34:26.730
is determinant of A times
determinant of B, and for me
00:34:26.730 --> 00:34:31.750
that one is like, that's
a very valuable property,
00:34:31.750 --> 00:34:34.449
and it's sort of like partly
coming out of the blue,
00:34:34.449 --> 00:34:37.050
because we haven't been
multiplying matrices
00:34:37.050 --> 00:34:41.159
and here suddenly
this determinant
00:34:41.159 --> 00:34:44.590
has this multiplying property.
00:34:44.590 --> 00:34:46.750
Remember, it didn't have
the linear property,
00:34:46.750 --> 00:34:48.810
it didn't have the
adding property.
00:34:48.810 --> 00:34:52.389
The determinant
of A plus B is not
00:34:52.389 --> 00:34:57.210
the sum of the determinants,
but the determinant of A times B
00:34:57.210 --> 00:35:01.220
is the product, is the
product of the determinants.
00:35:01.220 --> 00:35:06.955
OK, so for example, what's
the determinant of A inverse?
00:35:12.560 --> 00:35:14.240
Using that property nine.
00:35:19.360 --> 00:35:21.230
Let me, let me put
that under here
00:35:21.230 --> 00:35:27.800
because the camera is happier
if it can focus on both at once.
00:35:27.800 --> 00:35:29.140
So let me put it underneath.
00:35:29.140 --> 00:35:34.510
The determinant of A
inverse, because property ten
00:35:34.510 --> 00:35:40.420
will come in that space.
00:35:40.420 --> 00:35:44.530
What does this tell me about
A inverse, its determinant?
00:35:44.530 --> 00:35:47.730
OK, well, what do I
know about A inverse?
00:35:47.730 --> 00:35:54.980
I know that A inverse
times A is odd.
00:35:54.980 --> 00:35:55.960
So what I going to do?
00:35:59.200 --> 00:36:02.450
I'm going to take
determinants of both sides.
00:36:02.450 --> 00:36:06.290
The determinant of
I is one, and what's
00:36:06.290 --> 00:36:10.390
the determinant of A inverse A?
00:36:10.390 --> 00:36:13.670
That's a product of
two matrices, A and B.
00:36:13.670 --> 00:36:15.510
So it's the product
of the determinant,
00:36:15.510 --> 00:36:16.830
so what I learning?
00:36:16.830 --> 00:36:18.840
I'm learning that
the determinant
00:36:18.840 --> 00:36:24.070
of A inverse times
the determinant of A
00:36:24.070 --> 00:36:27.800
is the determinant of
I, that's this one.
00:36:27.800 --> 00:36:31.870
Again, I happily
use property one.
00:36:31.870 --> 00:36:35.950
OK, so that tells me that
the determinant of A inverse
00:36:35.950 --> 00:36:37.270
is one over.
00:36:37.270 --> 00:36:40.000
Here's my -- here's
my conclusion --
00:36:40.000 --> 00:36:53.910
is one over the
determinant of A.
00:36:53.910 --> 00:36:55.810
I guess that that --
00:36:55.810 --> 00:36:59.620
I, I always try to think, well,
do we know some cases of that?
00:36:59.620 --> 00:37:04.450
Of course, we know it's right
already if A is diagonal.
00:37:04.450 --> 00:37:09.000
If A is a diagonal matrix,
then its determinant
00:37:09.000 --> 00:37:10.480
is just a product
of those numbers.
00:37:10.480 --> 00:37:14.890
So if A is, for
example, two-three,
00:37:14.890 --> 00:37:20.740
then we know that A-inverse
is one-half one-third,
00:37:20.740 --> 00:37:26.440
and sure enough, that
has determinant six,
00:37:26.440 --> 00:37:29.430
and that has
determinant one-sixth.
00:37:29.430 --> 00:37:32.360
And our rule checks.
00:37:32.360 --> 00:37:39.490
So somehow this proof,
this property has to --
00:37:39.490 --> 00:37:41.960
somehow the proof
of that property --
00:37:41.960 --> 00:37:45.950
if we can boil it down to
diagonal matrices then we can
00:37:45.950 --> 00:37:49.140
read it off, whether
it's A and A-inverse,
00:37:49.140 --> 00:37:52.660
or two different diagonal
matrices A and B.
00:37:52.660 --> 00:37:54.430
For diagonal --
so what I saying?
00:37:54.430 --> 00:37:59.050
I'm saying for a
diagonal matrices, check.
00:37:59.050 --> 00:38:02.380
But we'd have to do
elimination steps,
00:38:02.380 --> 00:38:08.510
we'd have to patiently
do the, the, argument
00:38:08.510 --> 00:38:11.810
if we want to use these
previous properties to get it
00:38:11.810 --> 00:38:12.980
for other matrices.
00:38:12.980 --> 00:38:18.094
And it also tells me -- what,
just let's, see what else
00:38:18.094 --> 00:38:18.760
it's telling me.
00:38:18.760 --> 00:38:21.900
What's the determinant
of, of A-squared?
00:38:21.900 --> 00:38:26.750
If I take a matrix
and square it?
00:38:26.750 --> 00:38:30.460
Then the determinant
just got squared.
00:38:30.460 --> 00:38:31.140
Right?
00:38:31.140 --> 00:38:34.180
The determinant of
A-squared is the determinant
00:38:34.180 --> 00:38:35.864
of A times the determinant of A.
00:38:35.864 --> 00:38:38.030
So if I square the matrix,
I square the determinant.
00:38:38.030 --> 00:38:43.180
If I double the matrix, what do
I do to the non-zeroes flipped
00:38:43.180 --> 00:38:46.350
to the other side of the
diagonal, determinant?
00:38:46.350 --> 00:38:47.930
Think about that one.
00:38:47.930 --> 00:38:52.690
If I double the matrix, what
-- so the determinant of A,
00:38:52.690 --> 00:38:56.150
since I'm writing down,
like, facts that follow,
00:38:56.150 --> 00:39:02.500
the determinant of A-squared
is the determinant of A,
00:39:02.500 --> 00:39:04.740
all squared.
00:39:04.740 --> 00:39:09.580
The determinant of 2A is what?
00:39:12.250 --> 00:39:16.380
That's A plus A.
00:39:16.380 --> 00:39:19.910
But wait, er, I
don't want the answer
00:39:19.910 --> 00:39:22.410
to determinant of A here.
00:39:22.410 --> 00:39:23.150
That's wrong.
00:39:23.150 --> 00:39:25.860
It's not two determinant
of A, What is it?
00:39:25.860 --> 00:39:28.880
OK, one more coming,
which I I have to make,
00:39:28.880 --> 00:39:32.190
what's the number that I have
to multiply determinant of A
00:39:32.190 --> 00:39:34.900
by if I double the
whole matrix, if I
00:39:34.900 --> 00:39:36.710
double every entry
in the matrix?
00:39:36.710 --> 00:39:38.220
What happens to the determinant?
00:39:38.220 --> 00:39:41.230
If that were possible,
that would be a bad thing,
00:39:41.230 --> 00:39:43.640
Supposed it's an n-by-n
matrix. that gets --
00:39:43.640 --> 00:39:44.850
get down to triangular
00:39:44.850 --> 00:39:46.690
Two to the n, right.
00:39:46.690 --> 00:39:48.120
Two to the nth.
00:39:48.120 --> 00:39:50.650
Now, why is it two to the
nth, and not just two?
00:39:54.700 --> 00:39:58.070
So why is it two to the nth?
00:39:58.070 --> 00:40:02.230
Because I'm factoring
out two from every row.
00:40:02.230 --> 00:40:05.610
There's a factor -- this has
a factor two in every row,
00:40:05.610 --> 00:40:08.640
so I can factor two
out of the first row.
00:40:08.640 --> 00:40:12.120
I factor a different two out of
the second row, a different two
00:40:12.120 --> 00:40:15.640
out of the nth row, so I've
got all those twos coming out.
00:40:15.640 --> 00:40:20.290
So it's like volume,
really, and that's
00:40:20.290 --> 00:40:23.570
one of the great
applications of determinants.
00:40:23.570 --> 00:40:30.860
If I -- if I have a box
and I double all the sides,
00:40:30.860 --> 00:40:35.800
I multiply the volume
by two to the nth.
00:40:35.800 --> 00:40:38.440
If it's a box in
three dimensions,
00:40:38.440 --> 00:40:40.970
I multiply the volume by 8.
00:40:43.710 --> 00:40:47.190
So this is rule 3A here.
00:40:47.190 --> 00:40:49.480
This is rule nine.
00:40:49.480 --> 00:40:55.100
And notice the way this
rule sort of checks out with
00:40:55.100 --> 00:41:02.550
the singular/non-singular
stuff, that if A is invertible,
00:41:02.550 --> 00:41:05.650
what does that mean
about its determinant?
00:41:05.650 --> 00:41:08.020
It's not zero, and
therefore this makes sense.
00:41:10.770 --> 00:41:12.940
The case when
determinant of A is
00:41:12.940 --> 00:41:19.870
zero, that's the case where my
formula doesn't work anymore.
00:41:19.870 --> 00:41:24.030
If determinant of A is
zero, this is ridiculous,
00:41:24.030 --> 00:41:25.800
and that's ridiculous.
00:41:25.800 --> 00:41:31.300
A-inverse doesn't exist, and one
over zero doesn't make sense.
00:41:31.300 --> 00:41:36.090
So don't miss this property.
00:41:36.090 --> 00:41:38.560
It's sort of, like,
amazing that it can...
00:41:38.560 --> 00:41:44.660
And the tenth property is
equally simple to state,
00:41:44.660 --> 00:41:47.720
that the determinant
of A transposed
00:41:47.720 --> 00:41:57.010
equals the determinant of A.
00:41:57.010 --> 00:42:03.030
And of course, let's just
check it on the ab cd guy.
00:42:03.030 --> 00:42:07.075
We could check that sure
enough, that's ab cd, it works.
00:42:09.710 --> 00:42:14.350
It's ad - bc, it's
ad - bc, sure enough.
00:42:14.350 --> 00:42:19.140
So that transposing did
not change the determinant.
00:42:19.140 --> 00:42:24.790
But what it does change is --
00:42:24.790 --> 00:42:28.400
well, what it does is
it lists, so all --
00:42:28.400 --> 00:42:31.340
I've been working with rows.
00:42:31.340 --> 00:42:35.690
If a row is all zeroes,
the determinant is zero.
00:42:35.690 --> 00:42:40.360
But now, with rule
ten, I know what to do
00:42:40.360 --> 00:42:42.750
is a column is all zero.
00:42:42.750 --> 00:42:46.550
If a column is all zero,
what's the determinant?
00:42:46.550 --> 00:42:48.260
Zero, again.
00:42:48.260 --> 00:42:53.250
So, like all those properties
about rows, exchanging two rows
00:42:53.250 --> 00:42:55.080
reverses the sign.
00:42:55.080 --> 00:42:58.210
Now, exchanging two
columns reverses
00:42:58.210 --> 00:43:00.990
the sign, because
I can always, if I
00:43:00.990 --> 00:43:03.690
want to see why,
I can transpose,
00:43:03.690 --> 00:43:08.680
those columns become rows, I do
the exchange, I transpose back.
00:43:08.680 --> 00:43:11.760
And I've done a
column operation.
00:43:11.760 --> 00:43:17.530
So, in, in conclusion, there was
nothing special about row one,
00:43:17.530 --> 00:43:20.610
'cause I could exchange
rows, and now there's
00:43:20.610 --> 00:43:25.060
nothing special about rows that
isn't equally true for columns
00:43:25.060 --> 00:43:26.980
because this is the same.
00:43:26.980 --> 00:43:27.580
OK.
00:43:27.580 --> 00:43:32.080
And again, maybe I won't --
00:43:32.080 --> 00:43:33.320
oh, let's see.
00:43:33.320 --> 00:43:33.820
Do we...?
00:43:33.820 --> 00:43:37.930
Maybe it's worth seeing a
quick proof of this number ten,
00:43:37.930 --> 00:43:44.620
quick, quick, er,
proof of number ten.
00:43:44.620 --> 00:43:48.970
Er, let me take the
-- this is number ten.
00:43:48.970 --> 00:43:51.380
A transposed equals A.
00:43:51.380 --> 00:43:56.480
Determinate of A transposed
equals determinate of A.
00:43:56.480 --> 00:43:58.110
That's what I should have said.
00:43:58.110 --> 00:43:59.280
OK.
00:43:59.280 --> 00:44:07.820
So, let's just, er.
00:44:07.820 --> 00:44:11.450
A typical matrix A,
if I use elimination,
00:44:11.450 --> 00:44:16.100
this factors into LU.
00:44:16.100 --> 00:44:21.710
And the transpose is U
transpose, l transpose.
00:44:25.430 --> 00:44:26.370
Er... let me.
00:44:29.150 --> 00:44:36.870
So this is proof, this is
proof number ten, using --
00:44:36.870 --> 00:44:39.820
well, I don't know which
ones I'll use, so I'll put
00:44:39.820 --> 00:44:42.840
'em all in, one to nine.
00:44:42.840 --> 00:44:43.650
OK.
00:44:43.650 --> 00:44:47.400
I'm going to prove number
ten by using one to nine.
00:44:47.400 --> 00:44:50.910
I won't cover every case,
but I'll cover almost every
00:44:50.910 --> 00:44:51.550
case.
00:44:51.550 --> 00:44:55.400
So in almost every case,
A can factor into LU,
00:44:55.400 --> 00:44:57.710
and A transposed can
factor into that.
00:44:57.710 --> 00:45:00.070
Now, what do I do next?
00:45:00.070 --> 00:45:03.910
So I want to prove that
these are the same.
00:45:03.910 --> 00:45:06.610
I see a product here.
00:45:06.610 --> 00:45:09.560
So I use rule nine.
00:45:09.560 --> 00:45:14.860
So, now what I want to prove is,
so now I know that this is LU,
00:45:14.860 --> 00:45:19.810
this is U transposed
and l transposed.
00:45:19.810 --> 00:45:24.010
Now, just for a practice, what
are all those determinants?
00:45:24.010 --> 00:45:28.710
So this is, this is, this is
prove this, prove this, prove
00:45:28.710 --> 00:45:32.460
this, and now I'm
ready to do it.
00:45:32.460 --> 00:45:34.460
What's the determinant of l?
00:45:34.460 --> 00:45:40.240
You remember what l is, it's
this lower triangular matrix
00:45:40.240 --> 00:45:43.085
with ones on the diagonals.
00:45:43.085 --> 00:45:44.710
So what is the
determinant of that guy?
00:45:44.710 --> 00:45:45.210
I- It's one.
00:45:48.640 --> 00:45:53.000
Any time I have this
triangular matrix,
00:45:53.000 --> 00:46:00.390
I can get rid of
all the non-zeroes,
00:46:00.390 --> 00:46:07.920
down to the diagonal case.
00:46:07.920 --> 00:46:12.750
The determinate of l is one.
00:46:12.750 --> 00:46:21.810
How about the determinant
of l transposed?
00:46:21.810 --> 00:46:25.410
That's triangular also, right?
00:46:25.410 --> 00:46:28.050
Still got those ones
on the diagonal,
00:46:28.050 --> 00:46:59.050
it's just the matrices and then
get down to diagonal matrices.
00:46:59.050 --> 00:47:00.600
right?
00:47:00.600 --> 00:47:09.290
If If I could --
why would it be bad?
00:47:09.290 --> 00:47:11.800
My whole lecture
would die, right?
00:47:11.800 --> 00:47:37.360
Because rule two said that if
you do seven row exchanges,
00:47:37.360 --> 00:47:47.700
then the sign of the
determinant reverses.
00:47:47.700 --> 00:47:56.820
But if you do ten row exchanges,
the sign of the determinant
00:47:56.820 --> 00:48:02.520
stays the same, because minus
one ten times is plus one.
00:48:02.520 --> 00:48:16.630
OK, so there's a hidden
fact here, that every --
00:48:16.630 --> 00:48:19.150
like, every permutation,
the permutations
00:48:19.150 --> 00:48:23.310
are either odd or even.
00:48:23.310 --> 00:48:26.360
I could get the permutation
with seven row exchanges,
00:48:26.360 --> 00:48:27.770
then I could
probably get it with
00:48:27.770 --> 00:48:31.430
twenty-one, or twenty-three,
or a hundred and one,
00:48:31.430 --> 00:48:33.430
if it's an odd one.
00:48:33.430 --> 00:48:35.980
Or an even number
of permutations, so,
00:48:35.980 --> 00:48:39.070
but that's the key
fact that just takes
00:48:39.070 --> 00:48:43.760
another little
algebraic trick to see,
00:48:43.760 --> 00:48:45.947
and that means that the
determinant is well-defined
00:48:45.947 --> 00:48:47.530
by properties one,
two, three and it's
00:48:47.530 --> 00:48:47.580
got properties four to ten.
00:48:47.580 --> 00:48:47.660
OK, that's today
and I'll try to get
00:48:47.660 --> 00:48:47.890
the homework for next Wednesday
onto the web this afternoon.
00:48:47.890 --> 00:48:49.440
Thanks.