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OK, this lecture is like the
beginning of the second half of
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this course because up to now we
paid a lot of attention to
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rectangular matrices.
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Now, concentrating on square
matrices, so we're at two big
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topics.
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The determinant of a square
matrix, so this is the first
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lecture in that
new chapter on determinants,
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and the reason,
the big reason we need the
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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 with every
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square matrix,
so every square matrix has this
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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 the
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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
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information as possible into a
single number,
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and of course the one fact that
you've seen before and we have
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to see it again is the matrix is
invertible when the determinant
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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,
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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 as easy to tell
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you as that.
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But actually the second
property is pretty
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straightforward too,
and then once we get the third
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we will actually 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 exchange two
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rows of a matrix.
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What happens to the
determinant?
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So, property two is exchange
rows, reverse the sign of the
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determinant.
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A lot of plus and minus signs.
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I even wrote here,
"plus and minus
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signs," because this is,
like, that's what you have to
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pay attention to in the formulas
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 two rows to get a
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different matrix that reverses
the sign of the determinant.
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And so now, actually,
what matrices do we now know
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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 of
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every other matrix that comes
from row exchanges from the
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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
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exchanges 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.
if I could carry on this board,
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I could, like,
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 --
I mean, I'd better write
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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 -- 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 --
well, ultimately I want to get
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to, the formula that we all
know.
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Let me put that way over here,
that the determinant of a
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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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that we already know,
so that every property,
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I can, like,
check that it's correct for
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two-by-twos.
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But the whole point of these
properties is that they're going
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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 a test for
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invertibility and other great
properties 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
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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 -- I'm going to
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erase that.
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That's, like,
what I'm headed for but I'm not
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there yet.
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It's the one we know and you'll
see that it's checked out by
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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 and
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leave the other row or other n-1
rows alone, what happens to the
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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 -- I'm
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always keeping this second row
the same, or that last n-1 rows
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are all staying the same.
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I'm just working -- I'm just
looking inside the first
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row and if I have an a+a' there
and a b+b' there -- sorry,
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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
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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
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this 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 if all the
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other rows stay the same.
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I not saying that -- let me
emphasize.
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I not saying that the
determinant of A plus B is
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determinant of A plus
determinant of B.
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I not saying that.
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I'd better -- can I -- how do I
get it onto tape that I'm not
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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
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second row, if I had this --
but not, I can't,
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I can't have a combination in
both first and second.
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If I had a combination in the
second row, then I could use
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rule two to put it up in the
first row, use my property and
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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, and from those
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properties, so that's properties
one, two, three.
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From those, I want to get all
-- I'm going to learn a lot more
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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
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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
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check, 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
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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
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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,
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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,
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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 has to be zero from
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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 says that the sign
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did change.
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So the -- so I,
I have a determinant whose sign
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doesn't change and does change,
and the only possibility then
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is that the determinant is zero.
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You see the reasoning there?
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Straightforward.
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Property two just told us,
hey, if we've got two equal
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rows we.
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. --> 00:14:15
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we've got a zero determinant.
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And of course in our minds,
that matches the fact that if I
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have two equal rows the matrix
isn't invertible.
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If I have two equal rows,
I know that the rank is less
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than n.
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OK, ready for property five.
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Now, property five you'll
recognize as P.
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It says that the elimination
step that I'm always doing,
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subtract a multiple,
subtract some multiple l times
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row one from another row,
row k, let's say.
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You remember why I did that.
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In elimination I'm always
choosing
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00:15:14 --> 00:15:19
this multiplier so as to
produce zero in that position.
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Or row I from row k,
maybe I should just make very
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clear that there's nothing
special about row one here.
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OK, so that,
you can see why I want that
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one, because that will allow me
to start with this full matrix
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whose
determinant I don't know,
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and I can do elimination and
clean it out.
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I can get zeroes below the
diagonal by these elimination
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steps and the point is that the
determinant, the determinant
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00:16:03 --> 00:16:05
doesn't change.
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So all those steps of
elimination are OK not changing
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the determinant.
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00:16:15 --> 00:16:20
In our language in the early
chapter the determinant of A is
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going to be the same as the
determinant of U,
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the upper triangular one.
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It just has the pivots on the
diagonal.
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That's why we'll want this
property.
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OK, do you see where that
property's coming from?
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Let me do the two-by-two case.
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Let me do the two-by-two case
here.
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So, I'll keep property five
going along.
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So what I doing?
246
00:16:48 --> 00:16:53
I'm going to keep -- I'm going
to have ab cd,
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00:16:53 --> 00:16:59
but I'm going to subtract l
times the first row from the
248
00:16:59 --> 00:17:00
second row.
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And that's the determinant and
of
250
00:17:05 --> 00:17:09
course I can multiply that out
and figure out,
251
00:17:09 --> 00:17:13
sure enough,
ad-bc is there and this minus
252
00:17:13 --> 00:17:17
ALB plus ALB cancels out,
but I just cheated,
253
00:17:17 --> 00:17:17
right?
254
00:17:17 --> 00:17:20
I've got to use the properties.
255
00:17:20 --> 00:17:22
So what property?
256
00:17:22 --> 00:17:25
Well, of course,
this is a com -- I'm keeping
257
00:17:25 --> 00:17:30
the first row the same and the
second
258
00:17:30 --> 00:17:34
row has a c and a d,
and then there's the
259
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determinant of the A and the B,
and the minus LA,
260
00:17:39 --> 00:17:41
and the minus LB.
261
00:17:41 --> 00:17:44
So what property was that?
3B.
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00:17:44 --> 00:17:50
I kept one row the same and I
had a combination in the second,
263
00:17:50 --> 00:17:56
in the other row,
and I just separated it out.
264
00:17:56 --> 00:17:59
OK, so that's property 3.
265
00:17:59 --> 00:18:02
That's by number 3,
3B if you like.
266
00:18:02 --> 00:18:03
OK, now use 3A.
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00:18:03 --> 00:18:08
How do you use 3A,
which says I can factor out an
268
00:18:08 --> 00:18:12
l, I can factor out a minus l
here.
269
00:18:12 --> 00:18:17
I can factor a minus l out from
this row, no problem.
270
00:18:17 --> 00:18:18.52
That was 3A.
271
00:18:18.52 --> 00:18:26
So now I've used property three
and now I'm ready for the kill.
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00:18:26 --> 00:18:30
Property four says that this
determinant is zero,
273
00:18:30 --> 00:18:31
has two equal rows.
274
00:18:31 --> 00:18:34
You see how that would always
work?
275
00:18:34 --> 00:18:38
I subtract a multiple of one
row from another one.
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00:18:38 --> 00:18:44
It gives me a combination in
row k of the old row and l times
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00:18:44 --> 00:18:50
this copy of the higher row,
and then if -- since I have two
278
00:18:50 --> 00:18:56
equal rows, that's zero,
so the determinant after
279
00:18:56 --> 00:19:00
elimination is the same as
before.
280
00:19:00 --> 00:19:01
Good.
281
00:19:01 --> 00:19:01
OK.
282
00:19:01 --> 00:19:09.22
Now, let's see -- if I rescue
my glasses, I can see what's
283
00:19:09.22 --> 00:19:10
property six.
284
00:19:10 --> 00:19:15
Oh, six is easy,
plenty of space.
285
00:19:15 --> 00:19:23
Row of zeroes leads to
determinant of A equals zero.
286
00:19:23 --> 00:19:25
A complete row of zeroes.
287
00:19:25 --> 00:19:27
So I'm again,
this is like,
288
00:19:27 --> 00:19:31
something I'll use in the
singular case.
289
00:19:31 --> 00:19:36
Actually, you can look ahead to
why I need these properties.
290
00:19:36 --> 00:19:40
So I'm going to use property
five, the elimination,
291
00:19:40 --> 00:19:45
use this stuff to say that this
determinant is the same as that
292
00:19:45 --> 00:19:50
determinant and I'll
produce a zero there.
293
00:19:50 --> 00:19:52
But what if I also produce a
zero there?
294
00:19:52 --> 00:19:56
What if elimination gives a row
of zeroes?
295
00:19:56 --> 00:19:59
That, that used to be my test
for, mmm, singular,
296
00:19:59 --> 00:20:03
not invertible,
rank two -- rank less than N,
297
00:20:03 --> 00:20:06
and now I'm seeing it's also
gives determinant zero.
298
00:20:06 --> 00:20:11
How do I get that one from the
previous properties?
299
00:20:11 --> 00:20:17
'Cause I -- this is not a new
law, this has got to come from
300
00:20:17 --> 00:20:18
the old ones.
301
00:20:18 --> 00:20:20
So what shall I do?
302
00:20:20 --> 00:20:23
Yeah, use -- that's brilliant.
303
00:20:23 --> 00:20:26
If you use 3A with T equals
zero.
304
00:20:26 --> 00:20:27
Right.
305
00:20:27 --> 00:20:32
So I have this zero zero cd,
and I'm trying to show that
306
00:20:32 --> 00:20:35
that
determinant is zero.
307
00:20:35 --> 00:20:42
OK, so the zero is the same is
-- five, can I take T equals
308
00:20:42 --> 00:20:46
five, just to,
like, pin it down?
309
00:20:46 --> 00:20:49
I'll multiply this row by five.
310
00:20:49 --> 00:20:54
Five, well then,
five of this should -- if I,
311
00:20:54 --> 00:21:01
if there's a factor five in
that, you see what -- so this is
312
00:21:01 --> 00:21:06.52
property 3A, with taking T as
five.
313
00:21:06.52 --> 00:21:11
If I multiply a row by five,
out comes a five.
314
00:21:11 --> 00:21:13
So why I doing this?
315
00:21:13 --> 00:21:17
Oh, because that's still zero
zero, right?
316
00:21:17 --> 00:21:22.58
So that's this determinant
equals five times this
317
00:21:22.58 --> 00:21:27
determinant, and the determinant
has to be zero.
318
00:21:27 --> 00:21:31
I think I didn't do that the
very best way.
319
00:21:31 --> 00:21:37
You were, yeah,
you had the idea better.
320
00:21:37 --> 00:21:41
Multiply, yeah,
take T equals zero.
321
00:21:41 --> 00:21:43
Was that your idea?
322
00:21:43 --> 00:21:46
Take T equals zero in rule 3B.
323
00:21:46 --> 00:21:52.49
If T is zero in rule 3B,
and I bring the camera back to
324
00:21:52.49 --> 00:21:54
rule 3B -- sorry.
325
00:21:54 --> 00:21:59
If T is zero,
then I have a zero zero there
326
00:21:59 --> 00:22:03
and the determinant is zero.
327
00:22:03 --> 00:22:08
OK, one way or another,
a row of zeroes means zero
328
00:22:08 --> 00:22:09
determinant.
329
00:22:09 --> 00:22:13
OK, now I have to get serious.
330
00:22:13 --> 00:22:19
The next properties are the
ones that we're building up to.
331
00:22:19 --> 00:22:22
OK, so I can do elimination.
332
00:22:22 --> 00:22:28
I can reduce to a triangular
matrix and now what's the
333
00:22:28 --> 00:22:33
determinant of that triangular
matrix?
334
00:22:33 --> 00:22:37
Suppose, suppose I -- all
right, rule seven.
335
00:22:37 --> 00:22:41.75
So suppose my matrix is now
triangular.
336
00:22:41.75 --> 00:22:47
So it's got -- so I even give
these the names of the pivots,
337
00:22:47 --> 00:22:51
d1, d2, to dn,
and stuff is up here,
338
00:22:51 --> 00:22:58
I don't know what that is,
but what I do know is this is
339
00:22:58 --> 00:22:59.14
all zeroes.
340
00:22:59.14 --> 00:23:03
That's all zeroes,
and I would like to know the
341
00:23:03 --> 00:23:09
determinant, because elimination
will get me to this.
342
00:23:09 --> 00:23:13.84
So once I'm here,
what's the determinant then?
343
00:23:13.84 --> 00:23:18.26
Let me use an eraser to get
those,
344
00:23:18.26 --> 00:23:23
get that vertical bar again,
so that I'm taking the
345
00:23:23 --> 00:23:29
determinant of U so that,
so, what is the determinant of
346
00:23:29 --> 00:23:32
an upper triangular matrix?
347
00:23:32 --> 00:23:35
Do you know the answer?
348
00:23:35 --> 00:23:38.97
It's just the product of the
d's.
349
00:23:38.97 --> 00:23:45
The -- these things that I
don't even put in letters for,
350
00:23:45 --> 00:23:51
because they don't matter,
the determinant is d1 times d2
351
00:23:51 --> 00:23:53
times
dn.
352
00:23:53 --> 00:24:00
If I have a triangular matrix,
then the diagonal is all I have
353
00:24:00 --> 00:24:01
to work with.
354
00:24:01 --> 00:24:05
And that's, that's telling us
then.
355
00:24:05 --> 00:24:11
We now have the way that
MATLAB, any reasonable software,
356
00:24:11 --> 00:24:14
would compute a determinant.
357
00:24:14 --> 00:24:19
If I have a matrix of size a
hundred, the way I would
358
00:24:19 --> 00:24:25
actually compute its
determinant would be
359
00:24:25 --> 00:24:32
elimination, make it triangular,
multiply the pivots together,
360
00:24:32 --> 00:24:38
the product of the pivots,
the product of pivots.
361
00:24:38 --> 00:24:44
Now, there's always in
determinants a plus or minus
362
00:24:44 --> 00:24:46
sign to remember.
363
00:24:46 --> 00:24:51
Where -- where does that come
into this rule?
364
00:24:51 --> 00:24:57
Could it be,
could the determinant be minus
365
00:24:57 --> 00:24:59
the product of the pivots?
366
00:24:59 --> 00:25:02
The determinant is d1,
d2, to dn.
367
00:25:02 --> 00:25:03.74
No doubt about that.
368
00:25:03.74 --> 00:25:08.73
But to get to this triangular
form, we may have had to do a
369
00:25:08.73 --> 00:25:12
row exchange,
so, so this -- it's the product
370
00:25:12 --> 00:25:16
of the pivots if there were no
row exchanges.
371
00:25:16 --> 00:25:20
If, if SLU code,
the simple LU code,
372
00:25:20 --> 00:25:23
the square LU went right
through.
373
00:25:23 --> 00:25:29
If we had to do some row
exchanges, then we've got to
374
00:25:29 --> 00:25:31
watch plus or minus.
375
00:25:31 --> 00:25:35
OK, but though -- this law is
simply that.
376
00:25:35 --> 00:25:37
OK, how do I prove that?
377
00:25:37 --> 00:25:42
Let's see, let me suppose that
d's are not
378
00:25:42 --> 00:25:43
zeroes.
379
00:25:43 --> 00:25:45
The pivots are not zeroes.
380
00:25:45 --> 00:25:49
And tell me,
how do I show that none of this
381
00:25:49 --> 00:25:52.7
upper stuff makes any
difference?
382
00:25:52.7 --> 00:25:55
How do I get zeroes there?
383
00:25:55 --> 00:25:56
By elimination,
right?
384
00:25:56 --> 00:26:00.9
I just multiply this row by the
right number,
385
00:26:00.9 --> 00:26:03
subtract from that row,
kills that.
386
00:26:03 --> 00:26:08
I multiply this row by the
right
387
00:26:08 --> 00:26:11
number, kills that,
by the right number,
388
00:26:11 --> 00:26:12
kills that.
389
00:26:12 --> 00:26:16
Now, you kill every one of
these off-diagonal terms,
390
00:26:16 --> 00:26:20.6
no problem and I'm just left
with the diagonal.
391
00:26:20.6 --> 00:26:25
Well, strictly speaking,
I still have to figure out why
392
00:26:25 --> 00:26:29
is, for a diagonal matrix now,
why is that the right
393
00:26:29 --> 00:26:31
determinant?
394
00:26:31 --> 00:26:34
I mean, we sure hope it is,
but why?
395
00:26:34 --> 00:26:38
I have to go back to properties
one, two, three.
396
00:26:38 --> 00:26:43
Why is -- now that the matrix
is suddenly diagonal,
397
00:26:43 --> 00:26:47
how do I know that the
determinant is just a product of
398
00:26:47 --> 00:26:50
those diagonal entries?
399
00:26:50 --> 00:26:52
Well, what I going to use?
400
00:26:52 --> 00:26:56
I'm going to use property 3A,
is
401
00:26:56 --> 00:26:57.01
that right?
402
00:26:57.01 --> 00:26:59
I'll factor this,
I'll factor this,
403
00:26:59 --> 00:27:04
I'll factor that d1 out and
have one and have the first row
404
00:27:04 --> 00:27:05
will be that.
405
00:27:05 --> 00:27:10
And then I'll factor out the
d2, shall I shall I put the d2
406
00:27:10 --> 00:27:13
here, and the second row will
look like that,
407
00:27:13 --> 00:27:14
and so on.
408
00:27:14 --> 00:27:18
So I've factored out all the
d's and
409
00:27:18 --> 00:27:20
what I left with?
410
00:27:20 --> 00:27:21
The identity.
411
00:27:21 --> 00:27:24
And what rule do I finally get
to use?
412
00:27:24 --> 00:27:25
Rule one.
413
00:27:25 --> 00:27:30.59
Finally, this is the point
where rule one finally chips in
414
00:27:30.59 --> 00:27:35
and says that this determinant
is one, so it's the product of
415
00:27:35 --> 00:27:36
the d's.
416
00:27:36 --> 00:27:40
So this was rules five,
to do elimination,
417
00:27:40 --> 00:27:44.15
3A to factor out
the D's, and,
418
00:27:44.15 --> 00:27:46
and our best friend,
rule one.
419
00:27:46 --> 00:27:51
And possibly rule two,
the exchanges may have been
420
00:27:51 --> 00:27:52
needed also.
421
00:27:52 --> 00:27:53
OK.
422
00:27:53 --> 00:27:59
Now I guess I have to consider
also the case if some d is zero,
423
00:27:59 --> 00:28:04
because I was assuming I could
use the d's to clean out and
424
00:28:04 --> 00:28:09
make a diagonal,
but what if --
425
00:28:09 --> 00:28:13
what if one of those diagonal
entries is zero?
426
00:28:13 --> 00:28:19
Well, then with elimination we
know that we can get a row of
427
00:28:19 --> 00:28:24
zeroes, and for a row of zeroes
I'm using rule six,
428
00:28:24 --> 00:28:28.73
the determinant is zero,
and that's right.
429
00:28:28.73 --> 00:28:31.95
So I can check the singular
case.
430
00:28:31.95 --> 00:28:37
In fact, I can now get to the
key point that determinant of A
431
00:28:37 --> 00:28:41
is zero,
exactly when,
432
00:28:41 --> 00:28:44
exactly when A is singular.
433
00:28:44 --> 00:28:51
And otherwise is not singular,
so that the determinant is a
434
00:28:51 --> 00:28:56
fair test for invertibility or
non-invertibility.
435
00:28:56 --> 00:29:03
So, I must be close to that
because I can take any matrix
436
00:29:03 --> 00:29:04
and get there.
437
00:29:04 --> 00:29:08
Do I -- did I have anything to
say?
438
00:29:08 --> 00:29:13
The, the proofs,
it starts by saying by
439
00:29:13 --> 00:29:16
elimination go from A to U.
440
00:29:16 --> 00:29:17
Oh, yeah.
441
00:29:17 --> 00:29:22
Actually looks to me like I
don't -- haven't said anything
442
00:29:22 --> 00:29:26
brand-new here,
that, that really,
443
00:29:26 --> 00:29:30.32
I've got this,
because let's just remember the
444
00:29:30.32 --> 00:29:30
reason.
445
00:29:30 --> 00:29:35.4
By elimination,
I can go from the original A to
446
00:29:35.4 --> 00:29:35
U.
447
00:29:35 --> 00:29:40
Well, OK, now suppose the
matrix is
448
00:29:40 --> 00:29:41
singular.
449
00:29:41 --> 00:29:45
If the matrix is singular,
what happens?
450
00:29:45 --> 00:29:51
Then by elimination I get a row
of zeroes and therefore the
451
00:29:51 --> 00:29:53
determinant is zero.
452
00:29:53 --> 00:29:58
And if the matrix is not
singular, I don't get zero,
453
00:29:58 --> 00:30:03
so maybe -- do you want me to
put this, like,
454
00:30:03 --> 00:30:04.74
in two parts?
455
00:30:04.74 --> 00:30:09
The determinant of A is not
zero
456
00:30:09 --> 00:30:11
when A is invertible.
457
00:30:11 --> 00:30:17
Because I've already -- in
chapter two we figured out when
458
00:30:17 --> 00:30:19
is the matrix invertible.
459
00:30:19 --> 00:30:25.14
It's invertible when
elimination produces a full set
460
00:30:25.14 --> 00:30:29
of pivots and now,
and we now, we know the
461
00:30:29 --> 00:30:34
determinant is the product of
those non-zero numbers.
462
00:30:34 --> 00:30:38
So those are the two cases.
463
00:30:38 --> 00:30:43
If it's singular,
I go to a row of zeroes.
464
00:30:43 --> 00:30:48
If it's invertible,
I go to U and then to the
465
00:30:48 --> 00:30:53
diagonal D, and then which --
and then to d1,
466
00:30:53 --> 00:30:54
d2, up to dn.
467
00:30:54 --> 00:30:59
As the formula -- so we have a
formula now.
468
00:30:59 --> 00:31:05
We have a formula for the
determinant and it's actually a
469
00:31:05 --> 00:31:11
very much more practical formula
than the
470
00:31:11 --> 00:31:12
ad-bc formula.
471
00:31:12 --> 00:31:17
Is it correct,
maybe you should just -- let's
472
00:31:17 --> 00:31:18
just check that.
473
00:31:18 --> 00:31:19
Two-by-two.
474
00:31:19 --> 00:31:23
What are the pivots of a
two-by-two matrix?
475
00:31:23 --> 00:31:28
What does elimination do with a
two-by-two matrix?
476
00:31:28 --> 00:31:31
It -- there's the first pivot,
fine.
477
00:31:31 --> 00:31:33
What's the second pivot?
478
00:31:33 --> 00:31:39
We subtract,
so I'm putting it in this upper
479
00:31:39 --> 00:31:41
triangular form.
480
00:31:41 --> 00:31:46
What do I -- my multiplier is c
over a, right?
481
00:31:46 --> 00:31:53
I multiply that row by c over a
and I subtract to get that zero,
482
00:31:53 --> 00:31:57.63
and here I have d minus c over
a times b.
483
00:31:57.63 --> 00:32:02
That's the elimination on a
two-by-two.
484
00:32:02 --> 00:32:08
So I've finally discovered that
the determinant of this matrix
485
00:32:08 --> 00:32:13
-- I've got it from the
properties, not by knowing the
486
00:32:13 --> 00:32:17
answer from last year,
that the determinant of this
487
00:32:17 --> 00:32:23
two-by-two is the product of A
times that, and of course when I
488
00:32:23 --> 00:32:28
multiply A by that,
the product of that and that
489
00:32:28 --> 00:32:32
is ad minus,
the a is canceled,
490
00:32:32 --> 00:32:32
bc.
491
00:32:32 --> 00:32:36
So that's great,
provided a isn't zero.
492
00:32:36 --> 00:32:41
If a was zero,
that step wasn't allowed,
493
00:32:41 --> 00:32:43
zero wasn't a pivot.
494
00:32:43 --> 00:32:48
So that's what I -- I've
covered all the bases.
495
00:32:48 --> 00:32:56.58
I have to -- if a is zero,
then I have to do the exchange,
496
00:32:56.58 --> 00:33:02
and if the exchange doesn't
work, it's because a is
497
00:33:02 --> 00:33:03
singular.
498
00:33:03 --> 00:33:09
OK, those are -- those are the
direct properties of the
499
00:33:09 --> 00:33:10.59
determinant.
500
00:33:10.59 --> 00:33:14
And now, finally,
I've got two more,
501
00:33:14 --> 00:33:15.97
nine and ten.
502
00:33:15.97 --> 00:33:19
And that's -- I think you
can...
503
00:33:19 --> 00:33:25
Like, the ones
we've got here are totally
504
00:33:25 --> 00:33:31
connected with our elimination
process and whether pivots are
505
00:33:31 --> 00:33:36.46
available and whether we get a
row of zeroes.
506
00:33:36.46 --> 00:33:41
I think all that you can
swallow in one shot.
507
00:33:41 --> 00:33:45
Let me tell you properties nine
and ten.
508
00:33:45 --> 00:33:48
They're quick to write down.
509
00:33:48 --> 00:33:51
They're very,
very useful.
510
00:33:51 --> 00:33:57
So I'll just write them down
and use them.
511
00:33:57 --> 00:34:03.36
Property nine says that the
determinant of a product -- if I
512
00:34:03.36 --> 00:34:05
multiply two matrices.
513
00:34:05 --> 00:34:12
So if I multiply two matrices,
A and B, that the determinant
514
00:34:12 --> 00:34:18.2
of the product is determinant of
A times determinant of B,
515
00:34:18.2 --> 00:34:23
and for me that one is like,
that's a very valuable
516
00:34:23 --> 00:34:29
property, and it's sort of like
partly coming
517
00:34:29 --> 00:34:33
out of the blue,
because we haven't been
518
00:34:33 --> 00:34:38
multiplying matrices and here
suddenly this determinant has
519
00:34:38 --> 00:34:40
this multiplying property.
520
00:34:40 --> 00:34:44
Remember, it didn't have the
linear property,
521
00:34:44 --> 00:34:47
it didn't have the adding
property.
522
00:34:47 --> 00:34:54.67
The determinant of A plus B is
not the sum of the determinants,
523
00:34:54.67 --> 00:35:00.57
but the determinant of A times
B is the product,
524
00:35:00.57 --> 00:35:04
is the product of the
determinants.
525
00:35:04 --> 00:35:10
OK, so for example,
what's the determinant of A
526
00:35:10 --> 00:35:11
inverse?
527
00:35:11 --> 00:35:14
Using that property nine.
528
00:35:14 --> 00:35:21
Let me, let me put that under
here because the camera is
529
00:35:21 --> 00:35:26
happier if it can focus on both
at once.
530
00:35:26 --> 00:35:31
So let me put it underneath.
531
00:35:31 --> 00:35:37
The determinant of A inverse,
because property ten will come
532
00:35:37 --> 00:35:39
in that space.
533
00:35:39 --> 00:35:44
What does this tell me about A
inverse, its determinant?
534
00:35:44 --> 00:35:48
OK, well, what do I know about
A inverse?
535
00:35:48 --> 00:35:52
I know that A inverse times A
is odd.
536
00:35:52 --> 00:35:54
So what I going to do?
537
00:35:54 --> 00:35:59
I'm going to take determinants
of
538
00:35:59 --> 00:36:00
both sides.
539
00:36:00 --> 00:36:05.78
The determinant of I is one,
and what's the determinant of A
540
00:36:05.78 --> 00:36:06
inverse A?
541
00:36:06 --> 00:36:10
That's a product of two
matrices, A and B.
542
00:36:10 --> 00:36:16
So it's the product of the
determinant, so what I learning?
543
00:36:16 --> 00:36:20.85
I'm learning that the
determinant of A inverse times
544
00:36:20.85 --> 00:36:25
the determinant of A is the
determinant of
545
00:36:25 --> 00:36:28
I, that's this one.
546
00:36:28 --> 00:36:33
Again, I happily use property
one.
547
00:36:33 --> 00:36:41
OK, so that tells me that the
determinant of A inverse is one
548
00:36:41 --> 00:36:42
over.
549
00:36:42 --> 00:36:49
Here's my -- here's my
conclusion -- is one over the
550
00:36:49 --> 00:36:51
determinant of A.
551
00:36:51 --> 00:36:58
I guess that that --
I, I always try to think,
552
00:36:58 --> 00:37:01
well, do we know some cases of
that?
553
00:37:01 --> 00:37:06
Of course, we know it's right
already if A is diagonal.
554
00:37:06 --> 00:37:11
If A is a diagonal matrix,
then its determinant is just a
555
00:37:11 --> 00:37:13
product of those numbers.
556
00:37:13 --> 00:37:19
So if A is, for example,
two-three, then we know that
557
00:37:19 --> 00:37:24
A-inverse is one-half
one-third, and sure enough,
558
00:37:24 --> 00:37:28
that has determinant six,
and that has determinant
559
00:37:28 --> 00:37:29
one-sixth.
560
00:37:29 --> 00:37:31.84
And our rule checks.
561
00:37:31.84 --> 00:37:37
So somehow this proof,
this property has to -- somehow
562
00:37:37 --> 00:37:43
the proof of that property -- if
we can boil it down to diagonal
563
00:37:43 --> 00:37:47
matrices then we can read it
off,
564
00:37:47 --> 00:37:52
whether it's A and A-inverse,
or two different diagonal
565
00:37:52 --> 00:37:53
matrices A and B.
566
00:37:53 --> 00:37:56
For diagonal -- so what I
saying?
567
00:37:56 --> 00:37:59
I'm saying for a diagonal
matrices, check.
568
00:37:59 --> 00:38:04
But we'd have to do elimination
steps, we'd have to patiently do
569
00:38:04 --> 00:38:10
the, the, argument if we want to
use these previous properties to
570
00:38:10 --> 00:38:13
get it
for other matrices.
571
00:38:13 --> 00:38:18
And it also tells me -- what,
just let's, see what else it's
572
00:38:18 --> 00:38:19
telling me.
573
00:38:19 --> 00:38:23
What's the determinant of,
of A-squared?
574
00:38:23 --> 00:38:25
If I take a matrix and square
it?
575
00:38:25 --> 00:38:29
Then the determinant just got
squared.
576
00:38:29 --> 00:38:29
Right?
577
00:38:29 --> 00:38:33
The determinant of A-squared is
the
578
00:38:33 --> 00:38:37
determinant of A times the
determinant of A.
579
00:38:37 --> 00:38:42
So if I square the matrix,
I square the determinant.
580
00:38:42 --> 00:38:46
If I double the matrix,
what do I do to the
581
00:38:46 --> 00:38:47
determinant?
582
00:38:47 --> 00:38:49
Think about that one.
583
00:38:49 --> 00:38:54
If I double the matrix,
what -- so the determinant of
584
00:38:54 --> 00:38:59
A, since I'm writing down,
like, facts
585
00:38:59 --> 00:39:04
that follow,
the determinant of A-squared is
586
00:39:04 --> 00:39:08
the determinant of A,
all squared.
587
00:39:08 --> 00:39:11
The determinant of 2A is what?
588
00:39:11 --> 00:39:13
That's A plus A.
589
00:39:13 --> 00:39:18
But wait, er,
I don't want the answer to
590
00:39:18 --> 00:39:20.83
determinant of A here.
591
00:39:20.83 --> 00:39:22
That's wrong.
592
00:39:22 --> 00:39:28
It's not two determinant of A,
What is it?
593
00:39:28 --> 00:39:33.63
what's the number that I have
to multiply determinant of A by
594
00:39:33.63 --> 00:39:38
if I double the whole matrix,
if I double every entry in the
595
00:39:38 --> 00:39:39
matrix?
596
00:39:39 --> 00:39:42
What happens to the
determinant?
597
00:39:42 --> 00:39:45
Supposed it's an n-by-n matrix.
598
00:39:45 --> 00:39:46
Two to the n,
right.
599
00:39:46 --> 00:39:48
Two to the nth.
600
00:39:48 --> 00:39:53
Now, why is it two to the nth,
and not just two?
601
00:39:53 --> 00:39:56
So why is it two to the nth?
602
00:39:56 --> 00:39:59
Because I'm factoring out two
from every row.
603
00:39:59 --> 00:40:04
There's a factor -- this has a
factor two in every row,
604
00:40:04 --> 00:40:07
so I can factor two out of the
first row.
605
00:40:07 --> 00:40:12
I factor a different two out of
the second row,
606
00:40:12 --> 00:40:17.99
a different two out of the nth
row, so I've got all those twos
607
00:40:17.99 --> 00:40:19
coming out.
608
00:40:19 --> 00:40:23
So it's like volume,
really, and that's one of the
609
00:40:23 --> 00:40:26
great applications of
determinants.
610
00:40:26 --> 00:40:31
If I -- if I have a box and I
double all the sides,
611
00:40:31 --> 00:40:36
I multiply the volume by two to
the nth.
612
00:40:36 --> 00:40:42
If it's a box in three
dimensions, I multiply the
613
00:40:42 --> 00:40:43
volume by 8.
614
00:40:43 --> 00:40:46
So this is rule 3A here.
615
00:40:46 --> 00:40:48
This is rule nine.
616
00:40:48 --> 00:40:55
And notice the way this rule
sort of checks out with the
617
00:40:55 --> 00:41:01
singular/non-singular stuff,
that if A is invertible,
618
00:41:01 --> 00:41:06
what does that mean about its
determinant?
619
00:41:06 --> 00:41:11
It's not zero,
and therefore this makes sense.
620
00:41:11 --> 00:41:17
The case when determinant of A
is zero, that's the case where
621
00:41:17 --> 00:41:19
my formula doesn't work anymore.
622
00:41:19 --> 00:41:23
If determinant of A is zero,
this is ridiculous,
623
00:41:23 --> 00:41:25
and that's ridiculous.
624
00:41:25 --> 00:41:30
A-inverse doesn't exist,
and one over zero doesn't make
625
00:41:30 --> 00:41:31.13
sense.
626
00:41:31.13 --> 00:41:34
So don't miss this property.
627
00:41:34 --> 00:41:39
It's sort of,
like, amazing that it can...
628
00:41:39 --> 00:41:46
And the tenth property is
equally simple to state,
629
00:41:46 --> 00:41:52
that the determinant of A
transposed equals the
630
00:41:52 --> 00:41:54
determinant of A.
631
00:41:54 --> 00:42:00
And of course,
let's just check it on the ab
632
00:42:00 --> 00:42:01
cd guy.
633
00:42:01 --> 00:42:06
We could check that sure
enough, that's ab cd,
634
00:42:06 --> 00:42:09
it works.
635
00:42:09 --> 00:42:12
It's ad - bc,
it's ad - bc,
636
00:42:12 --> 00:42:14
sure enough.
637
00:42:14 --> 00:42:19
So that transposing did not
change the determinant.
638
00:42:19 --> 00:42:26
But what it does change is --
well, what it does is it lists,
639
00:42:26 --> 00:42:30
so all -- I've been working
with rows.
640
00:42:30 --> 00:42:37
If a row is all zeroes,
the determinant is zero.
641
00:42:37 --> 00:42:41
But now, with rule ten,
I know what to do is a column
642
00:42:41 --> 00:42:42
is all zero.
643
00:42:42 --> 00:42:46
If a column is all zero,
what's the determinant?
644
00:42:46 --> 00:42:47
Zero, again.
645
00:42:47 --> 00:42:52
So, like all those properties
about rows, exchanging two rows
646
00:42:52 --> 00:42:53
reverses the sign.
647
00:42:53 --> 00:42:57
Now, exchanging two columns
reverses the sign,
648
00:42:57 --> 00:43:02
because I can always,
if I want to see why,
649
00:43:02 --> 00:43:05
I can transpose,
those columns become rows,
650
00:43:05 --> 00:43:08
I do the exchange,
I transpose back.
651
00:43:08 --> 00:43:11
And I've done a column
operation.
652
00:43:11 --> 00:43:15
So, in, in conclusion,
there was nothing special about
653
00:43:15 --> 00:43:20
row one, 'cause I could exchange
rows, and now there's nothing
654
00:43:20 --> 00:43:25.09
special about rows that isn't
equally true for
655
00:43:25.09 --> 00:43:28
columns because this is the
same.
656
00:43:28 --> 00:43:28
OK.
657
00:43:28 --> 00:43:33
And again, maybe I won't -- oh,
let's see.
658
00:43:33 --> 00:43:34
Do we...?
659
00:43:34 --> 00:43:40
Maybe it's worth seeing a quick
proof of this number ten,
660
00:43:40 --> 00:43:44
quick, quick,
er, proof of number ten.
661
00:43:44 --> 00:43:48
Er, let me take the -- this is
number ten.
662
00:43:48 --> 00:43:52
A transposed
equals A.
663
00:43:52 --> 00:43:59.99
Determinate of A transposed
equals determinate of A.
664
00:43:59.99 --> 00:44:04
That's what I should have said.
665
00:44:04 --> 00:44:05
OK.
666
00:44:05 --> 00:44:07
So, let's just,
er.
667
00:44:07 --> 00:44:13
A typical matrix A,
if I use elimination,
668
00:44:13 --> 00:44:16
this factors into LU.
669
00:44:16 --> 00:44:23
And the transpose is U
transpose, l transpose.
670
00:44:23 --> 00:44:25.21
Er...
let me.
671
00:44:25.21 --> .
672
. --> 00:44:30
this is to prove.
673
00:44:30 --> 00:44:33
So this is proof,
this is proof number ten,
674
00:44:33 --> 00:44:37
using -- well,
I don't know which ones I'll
675
00:44:37 --> 00:44:40
use, so I'll put 'em all in,
one to nine.
676
00:44:40 --> 00:44:40
OK.
677
00:44:40 --> 00:44:44
I'm going to prove number ten
by using one to nine.
678
00:44:44 --> 00:44:49
I won't cover every case,
but I'll cover almost every
679
00:44:49 --> 00:44:49
case.
680
00:44:49 --> 00:44:52
So in almost every case,
A can
681
00:44:52 --> 00:44:57
factor into LU,
and A transposed can factor
682
00:44:57 --> 00:44:58
into that.
683
00:44:58 --> 00:45:00
Now, what do I do next?
684
00:45:00 --> 00:45:04
So I want to prove that these
are the same.
685
00:45:04 --> 00:45:06
I see a product here.
686
00:45:06 --> 00:45:08
So I use rule nine.
687
00:45:08 --> 00:45:14
So, now what I want to prove
is, so now I know that this is
688
00:45:14 --> 00:45:19
LU, this is U transposed and l
transposed.
689
00:45:19 --> 00:45:23
Now, just for a practice,
what are all those
690
00:45:23 --> 00:45:25
determinants?
691
00:45:25 --> 00:45:28
So this is, this is,
this is prove this,
692
00:45:28 --> 00:45:33
prove this, prove this,
and now I'm ready to do it.
693
00:45:33 --> 00:45:35
What's the determinant of l?
694
00:45:35 --> 00:45:39
You remember what l is,
it's this lower triangular
695
00:45:39 --> 00:45:44
matrix with ones on the
diagonals.
696
00:45:44 --> 00:45:47
So what is the determinant of
that guy?
697
00:45:47 --> 00:45:48
I- It's one.
698
00:45:48 --> 00:45:53
Any time I have this triangular
matrix, I can get rid of all the
699
00:45:53 --> 00:45:56.98
non-zeroes, down to the diagonal
case.
700
00:45:56.98 --> 00:45:59
The determinate of l is one.
701
00:45:59 --> 00:46:02
How about the determinant of l
transposed?
702
00:46:02 --> 00:46:06.5
That's triangular also,
right?
703
00:46:06.5 --> 00:38:45
Still got those ones on the
diagonal, it's just the
704
00:38:45 --> 00:31:07
non-zeroes flipped to the other
side of the diagonal,
705
00:31:07 --> 00:26:43
but they didn't matter anyway.
706
00:26:43 --> 00:20:33
That's my proof,
really, that once I've got
707
00:20:33 --> 00:14:49
triangular matrices,
l and l transposed,
708
00:14:49 --> 00:06:27
or U and U transposed,
when they're triangular,4
709
00:06:27 --> 00:15:30
I'm down to the product of the
diagonal and if I transpose,
710
00:15:30 --> 00:17:04
who cares?
711
00:17:04 --> 00:24:43
OK, that's not -- I didn't put
in every comma and,
712
00:24:43 --> 00:33:00
and cross every T in that
proof, but that's really the
713
00:33:00 --> 00:33:57
proof.
714
00:33:57 --> 00:39:44
That's the, like,
concrete proof that,
715
00:39:44 --> 00:46:55
that gets -- get down to
triangular
716
00:46:55 --> 00:39:25
matrices and then get down to
diagonal matrices.
717
00:39:25 --> 00:32:33
OK, one more coming,
which I I have to make,
718
00:32:33 --> 00:23:36
because all math professors
watching this will be waiting
719
00:23:36 --> 00:22:29
for it.
720
00:22:29 --> 00:15:18
OK, so they had to wait until
the last minute.
721
00:15:18 --> 00:07:20
What I -- way,
way back in property two,4
722
00:07:20 --> 00:14:29.14
I said that if you do a row
exchange, the determinant
723
00:14:29.14 --> 00:16:16
changes sign.
724
00:16:16 --> 00:24:31
So if I do seven row exchanges,
the determinant changes sign,
725
00:24:31 --> 00:32:38
but it -- would it be possible
t- to produce the same matrix
726
00:32:38 --> 00:39:39
with seven row exchanges and
with ten row exchanges?
727
00:39:39 --> 00:47:46
If that were possible,
that would be a bad thing,
728
00:47:46 --> 00:47:46
right?
729
00:47:46 --> 00:47:49
If If I could -- why would it
be bad?
730
00:47:49 --> 00:47:52
My whole lecture would die,
right?
731
00:47:52 --> 00:47:56
Because rule two said that if
you do seven row exchanges,
732
00:47:56 --> 00:48:00
then the sign of the
determinant reverses.
733
00:48:00 --> 00:48:04
But if you do ten row
exchanges, the sign of the
734
00:48:04 --> 00:48:08
determinant stays the same,
because minus one
735
00:48:08 --> 00:48:10
ten times is plus one.
736
00:48:10 --> 00:48:15
OK, so there's a hidden fact
here, that every -- like,
737
00:48:15 --> 00:48:19
every permutation,
the permutations are either odd
738
00:48:19 --> 00:48:20
or even.
739
00:48:20 --> 00:48:25
I could get the permutation
with seven row exchanges,
740
00:48:25 --> 00:48:29
then I could probably get it
with twenty-one,
741
00:48:29 --> 00:48:32
or twenty-three,
or a hundred and one,
742
00:48:32 --> 00:48:34
if it's an odd one.
743
00:48:34 --> 00:48:36
Or an even number of
permutations,
744
00:48:36 --> 00:48:41
so, but that's the key fact
that just takes another little
745
00:48:41 --> 00:48:44
algebraic trick to see,
and that means that the
746
00:48:44 --> 00:48:48
determinant is well-defined by
properties one,
747
00:48:48 --> 00:48:51
two, three and it's got
properties
748
00:48:51 --> 00:48:52
four to ten.
749
00:48:52 --> 00:48:56
OK, that's today and I'll try
to get the homework for next
750
00:48:56 --> 00:48:59
Wednesday onto the web this
afternoon.
751
00:48:59 --> 00:49:02
Thanks.