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00:00:07,730 --> 00:00:11,436
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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00:00:22,240 --> 00:00:27,020
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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00:00:38,940 --> 00:00:43,030
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,
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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
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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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00:04:55,020 --> 00:04:58,730
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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00:05:16,720 --> 00:05:19,100
So this is the determinant
of a permutation.
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00:05:19,100 --> 00:05:22,720
Now, P is back to
standing for permutation.
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OK.
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00:05:23,560 --> 00:05:28,180
if I could carry on this
board, I could, like,
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do the two-by-two's.
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00:05:29,940 --> 00:05:34,790
So, property one tells me
that this two-by-two matrix.
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00:05:34,790 --> 00:05:37,560
Oh, I better write absolute --
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00:05:37,560 --> 00:05:41,150
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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00:05:51,700 --> 00:05:59,130
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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00:06:16,600 --> 00:06:23,190
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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00:09:02,390 --> 00:09:06,730
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,
168
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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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00:11:47,310 --> 00:11:58,720
Property four is if two rows are
equal, the determinant is zero.
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00:11:58,720 --> 00:12:02,720
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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00:12:14,280 --> 00:12:19,360
Now, of course I can -- in the
two-by-two case I can check,
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00:12:19,360 --> 00:12:23,980
sure, the determinant
of ab ab comes out zero.
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00:12:23,980 --> 00:12:28,950
But I want to see why
it's true for n-by-n.
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00:12:28,950 --> 00:12:36,100
Suppose row one equals row three
for a seven-by-seven matrix.
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00:12:36,100 --> 00:12:39,070
So two rows are the
same in a big matrix.
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00:12:39,070 --> 00:12:43,350
And all I have to work
with is these properties.
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00:12:43,350 --> 00:12:47,540
The exchange property,
which flips the sign,
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00:12:47,540 --> 00:12:53,580
and the linearity property which
works in each row separately.
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00:12:53,580 --> 00:12:57,170
OK, can you see the reason?
192
00:12:57,170 --> 00:13:02,200
How do I get this one out of
properties one, two, three?
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00:13:02,200 --> 00:13:04,680
Because -- that's all
I have to work with.
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00:13:04,680 --> 00:13:07,770
Everything has to come from
properties one, two, three.
195
00:13:07,770 --> 00:13:12,040
OK, so suppose I have a
matrix, and two rows are even.
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00:13:14,770 --> 00:13:16,310
How do I see that
its determinant
197
00:13:16,310 --> 00:13:22,120
has to be zero from
these properties?
198
00:13:22,120 --> 00:13:24,490
I do an exchange.
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00:13:24,490 --> 00:13:27,190
Property two is just
set up for this.
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00:13:27,190 --> 00:13:28,610
Use property two.
201
00:13:28,610 --> 00:13:33,090
Use exchange -- exchange rows.
202
00:13:33,090 --> 00:13:40,720
Exchange those rows, and
I get the same matrix.
203
00:13:40,720 --> 00:13:42,700
Of course, because
those rows were equal.
204
00:13:47,870 --> 00:13:50,580
So the determinant
didn't change.
205
00:13:50,580 --> 00:13:52,360
But on the other
hand, property two
206
00:13:52,360 --> 00:13:56,170
says that the sign did change.
207
00:13:56,170 --> 00:13:59,350
So the -- so I, I have
a determinant whose sign
208
00:13:59,350 --> 00:14:03,390
doesn't change and does change,
and the only possibility then
209
00:14:03,390 --> 00:14:06,520
is that the determinant is zero.
210
00:14:06,520 --> 00:14:08,780
You see the reasoning there?
211
00:14:08,780 --> 00:14:09,550
Straightforward.
212
00:14:09,550 --> 00:14:15,250
Property two just told us, hey,
if we've got two equal rows we.
213
00:14:15,250 --> 00:14:19,210
we've got a zero determinant.
214
00:14:19,210 --> 00:14:22,140
And of course in our minds,
that matches the fact
215
00:14:22,140 --> 00:14:26,550
that if I have two equal rows
the matrix isn't invertible.
216
00:14:26,550 --> 00:14:29,430
If I have two equal rows,
I know that the rank
217
00:14:29,430 --> 00:14:31,190
is less changes sign. than n.
218
00:14:31,190 --> 00:14:34,540
OK, ready for property five.
219
00:14:34,540 --> 00:14:38,900
Now, property five
you'll recognize as P.
220
00:14:38,900 --> 00:14:45,710
It says that the elimination
step that I'm always doing,
221
00:14:45,710 --> 00:14:51,850
or U and U transposed, when
they're triangular,4 subtract
222
00:14:51,850 --> 00:15:00,560
a multiple, subtract some
multiple l times row one from
223
00:15:00,560 --> 00:15:04,800
another row, row k, let's say.
224
00:15:07,950 --> 00:15:10,760
You remember why I did that.
225
00:15:10,760 --> 00:15:14,620
In elimination I'm always
choosing this multiplier so as
226
00:15:14,620 --> 00:15:17,190
to produce zero
in that position.
227
00:15:17,190 --> 00:15:21,050
What I -- way, way
back in property two,4
228
00:15:21,050 --> 00:15:24,230
Or row I from row k,
maybe I should just
229
00:15:24,230 --> 00:15:28,250
make very clear that there's
nothing special about row one
230
00:15:28,250 --> 00:15:30,530
here.
231
00:15:30,530 --> 00:15:34,400
OK, so that, you can see
why I want that who cares?
232
00:15:34,400 --> 00:15:37,350
one, because that
will allow me to start
233
00:15:37,350 --> 00:15:40,890
with this full matrix whose
determinant I don't know,
234
00:15:40,890 --> 00:15:45,590
and I can do elimination
and clean it out.
235
00:15:45,590 --> 00:15:47,830
I can get zeroes
below the diagonal
236
00:15:47,830 --> 00:15:50,970
by these elimination
steps and the point
237
00:15:50,970 --> 00:15:58,420
is that the determinant, the
determinant doesn't change.
238
00:16:10,210 --> 00:16:12,450
So all those steps
of elimination
239
00:16:12,450 --> 00:16:15,070
are OK not changing
the determinant.
240
00:16:15,070 --> 00:16:18,580
In our language in the early
chapter the determinant of A is
241
00:16:18,580 --> 00:16:20,900
So if I do seven row
exchanges, the determinant
242
00:16:20,900 --> 00:16:23,989
changes sign, going to be the
same as the determinant of U,
243
00:16:23,989 --> 00:16:25,030
the upper triangular one.
244
00:16:25,030 --> 00:16:27,090
It just has the pivots
on the diagonal.
245
00:16:27,090 --> 00:16:29,100
That's why we'll
want this property.
246
00:16:29,100 --> 00:16:31,250
OK, do you see where that
property's coming from?
247
00:16:34,180 --> 00:16:37,110
Let me do the two-by-two case.
248
00:16:37,110 --> 00:16:39,490
Let me do the
two-by-two case here.
249
00:16:39,490 --> 00:16:44,340
So, I'll keep property
five going along.
250
00:16:44,340 --> 00:16:45,300
So what I doing?
251
00:16:45,300 --> 00:16:46,890
I'm going to keep --
252
00:16:46,890 --> 00:16:52,910
I'm going to have
ab cd, but I'm going
253
00:16:52,910 --> 00:16:57,460
to subtract l times the first
row from the second row.
254
00:16:57,460 --> 00:17:03,182
And that's the
determinant and of
255
00:17:03,182 --> 00:17:03,890
OK, that's not --
256
00:17:03,890 --> 00:17:06,260
I didn't put in every
comma and, course
257
00:17:06,260 --> 00:17:10,900
I can multiply that out and
figure out, sure enough, ad-bc
258
00:17:10,900 --> 00:17:17,849
is there and this minus
ALB plus ALB cancels out,
259
00:17:17,849 --> 00:17:19,569
but I just cheated,
260
00:17:19,569 --> 00:17:20,190
right?
261
00:17:20,190 --> 00:17:21,760
I've got to use the properties.
262
00:17:21,760 --> 00:17:22,510
So what property?
263
00:17:22,510 --> 00:17:24,599
Well, of course,
this is a com --
264
00:17:24,599 --> 00:17:28,420
I'm keeping the first row
the same and the second row
265
00:17:28,420 --> 00:17:31,690
has a c and a d,
and then there's
266
00:17:31,690 --> 00:17:36,190
the determinant of the A
and the B, and the minus LA,
267
00:17:36,190 --> 00:17:37,240
and the minus LB.
268
00:17:41,840 --> 00:17:44,150
So what property was that?
269
00:17:44,150 --> 00:17:46,420
3B.
270
00:17:46,420 --> 00:17:49,340
I kept one row
the same and I had
271
00:17:49,340 --> 00:17:52,690
a combination in the
second, in the other row,
272
00:17:52,690 --> 00:17:56,520
and I just separated it out.
273
00:17:56,520 --> 00:17:59,300
OK, so that's property 3.
274
00:17:59,300 --> 00:18:03,350
That's by number
3, 3B if you like.
275
00:18:03,350 --> 00:18:04,980
OK, now use 3A.
276
00:18:04,980 --> 00:18:10,480
How do you use 3A, which
says I can factor out an l,
277
00:18:10,480 --> 00:18:13,110
I can factor out a minus l here.
278
00:18:13,110 --> 00:18:17,140
I can factor a minus l out
from this row, no problem.
279
00:18:17,140 --> 00:18:19,000
That was 3A.
280
00:18:19,000 --> 00:18:25,390
So now I've used property three
and now I'm ready for the kill.
281
00:18:25,390 --> 00:18:32,070
Property four says that
this determinant is zero,
282
00:18:32,070 --> 00:18:34,810
has two equal rows.
283
00:18:34,810 --> 00:18:37,100
You see how that
would always work?
284
00:18:37,100 --> 00:18:40,280
I subtract a multiple of
one row from another one.
285
00:18:40,280 --> 00:18:46,860
It gives me a combination in
row k of the old row and l times
286
00:18:46,860 --> 00:18:51,310
this copy of the higher
row, and then if --
287
00:18:51,310 --> 00:18:53,550
since I have two equal
rows, that's zero,
288
00:18:53,550 --> 00:18:56,990
so the determinant after
elimination is the same
289
00:18:56,990 --> 00:18:58,280
as before.
290
00:18:58,280 --> 00:18:59,430
Good.
291
00:18:59,430 --> 00:19:00,310
OK.
292
00:19:00,310 --> 00:19:04,170
Now, let's see -- if
I rescue my glasses,
293
00:19:04,170 --> 00:19:07,140
I can see what's property six.
294
00:19:07,140 --> 00:19:11,700
Oh, six is easy,
plenty of space.
295
00:19:11,700 --> 00:19:22,450
Row of zeroes leads to
determinant of A equals zero.
296
00:19:26,840 --> 00:19:28,380
A complete row of zeroes.
297
00:19:28,380 --> 00:19:32,550
So I'm again, this
is like, something
298
00:19:32,550 --> 00:19:34,880
I'll use in the singular case.
299
00:19:34,880 --> 00:19:39,380
Actually, you can look ahead
to why I need these properties.
300
00:19:39,380 --> 00:19:42,220
So I'm going to use property
five, the elimination,
301
00:19:42,220 --> 00:19:45,980
use this stuff to say
that this determinant is
302
00:19:45,980 --> 00:19:49,600
the same as that determinant
and I'll produce a zero there.
303
00:19:49,600 --> 00:19:51,910
But what if I also
produce a zero there?
304
00:19:51,910 --> 00:19:54,620
What if elimination
gives a row of zeroes?
305
00:19:54,620 --> 00:19:59,120
That, that used to be my
test for, mmm, singular,
306
00:19:59,120 --> 00:20:03,220
not invertible, rank
two -- rank less than N,
307
00:20:03,220 --> 00:20:07,120
and now I'm seeing it's
also gives determinant zero.
308
00:20:07,120 --> 00:20:12,600
How do I get that one from
the previous properties?
309
00:20:12,600 --> 00:20:15,006
'Cause I -- this
is not a new law,
310
00:20:15,006 --> 00:20:16,630
this has got to come
from the old ones.
311
00:20:16,630 --> 00:20:20,635
So what shall I do?
312
00:20:23,210 --> 00:20:24,690
Yeah, use -- that's brilliant.
313
00:20:24,690 --> 00:20:26,430
If you use 3A with
T equals zero.
314
00:20:26,430 --> 00:20:27,350
Right.
315
00:20:27,350 --> 00:20:32,760
So I have this zero
zero cd, and I'm
316
00:20:32,760 --> 00:20:35,850
trying to show that that
determinant is zero. triangular
317
00:20:35,850 --> 00:20:37,560
matrices, l and l transposed,
318
00:20:37,560 --> 00:20:41,000
OK, so the zero is
the same is -- five,
319
00:20:41,000 --> 00:20:45,900
can I take T equals five,
just to, like, pin it down?
320
00:20:45,900 --> 00:20:48,530
I'll multiply this row by five.
321
00:20:48,530 --> 00:20:52,780
Five, well then, five
of this should -- if I,
322
00:20:52,780 --> 00:21:01,300
if there's a factor five
in that, you see what --
323
00:21:01,300 --> 00:21:05,500
so this is property 3A,
with taking T as five.
324
00:21:05,500 --> 00:21:08,660
If I multiply a row by
five, out comes a five.
325
00:21:08,660 --> 00:21:14,450
So why I doing this?
326
00:21:14,450 --> 00:21:19,320
Oh, because that's
still zero zero, right?
327
00:21:19,320 --> 00:21:21,000
So that's this
determinant equals
328
00:21:21,000 --> 00:21:28,780
five times this determinant, and
the determinant has to be zero.
329
00:21:28,780 --> 00:21:34,100
I think I didn't do
that the very best way.
330
00:21:34,100 --> 00:21:36,670
You were, yeah, you
had the idea better.
331
00:21:36,670 --> 00:21:40,620
Multiply, yeah,
take T equals zero.
332
00:21:40,620 --> 00:21:44,170
Was that your idea?
333
00:21:44,170 --> 00:21:46,840
Take T equals zero in rule 3B.
334
00:21:46,840 --> 00:21:51,990
If T is zero in rule 3B, and I
bring the camera back to rule
335
00:21:51,990 --> 00:21:52,720
3B --
336
00:21:52,720 --> 00:21:55,260
sorry.
337
00:21:55,260 --> 00:22:01,300
If T is zero, then I
have a zero zero there
338
00:22:01,300 --> 00:22:03,330
and the determinant is zero.
339
00:22:03,330 --> 00:22:08,710
OK, one way or another, a row of
zeroes means zero determinant.
340
00:22:08,710 --> 00:22:14,930
OK, now I have to get serious.
341
00:22:14,930 --> 00:22:20,500
The next properties are the
ones that we're building up to.
342
00:22:20,500 --> 00:22:23,750
OK, so I can do elimination.
343
00:22:23,750 --> 00:22:26,540
I can reduce to a
triangular matrix
344
00:22:26,540 --> 00:22:30,120
and now what's the determinant
of that triangular matrix?
345
00:22:30,120 --> 00:22:34,100
OK, so they had to wait
until the last minute.
346
00:22:34,100 --> 00:22:37,280
Suppose, suppose I --
all right, rule seven.
347
00:22:37,280 --> 00:22:42,430
So suppose my matrix
is now triangular.
348
00:22:42,430 --> 00:22:44,660
So it's got --
349
00:22:44,660 --> 00:22:48,870
so I even give these the names
of the pivots, d1, d2, to dn,
350
00:22:48,870 --> 00:22:54,710
and stuff is up here, I
don't know what that is,
351
00:22:54,710 --> 00:22:57,970
but what I do know is
this is all zeroes.
352
00:22:57,970 --> 00:23:04,750
That's all zeroes, and I would
like to know the determinant,
353
00:23:04,750 --> 00:23:08,370
because elimination
will get me to this.
354
00:23:08,370 --> 00:23:11,610
So once I'm here, what's
the determinant then?
355
00:23:11,610 --> 00:23:16,560
Let me use an eraser to get
those, get that vertical bar
356
00:23:16,560 --> 00:23:22,350
again, so that I'm taking the
determinant of U so that, so,
357
00:23:22,350 --> 00:23:28,510
what is the determinant of
an upper triangular matrix?
358
00:23:28,510 --> 00:23:33,215
Do you know the answer?
359
00:23:36,090 --> 00:23:40,220
It's just the product
of the d's. for it.
360
00:23:40,220 --> 00:23:44,740
The -- these things that I
don't even put in letters
361
00:23:44,740 --> 00:23:53,020
for, because they don't matter,
the determinant is d1 times d2
362
00:23:53,020 --> 00:23:54,197
times dn.
363
00:23:57,120 --> 00:24:02,100
If I have a triangular
matrix, then the diagonal
364
00:24:02,100 --> 00:24:04,770
is all I have to work with.
365
00:24:04,770 --> 00:24:06,550
And that's, that's
telling us then.
366
00:24:06,550 --> 00:24:15,120
We now have the way that
MATLAB, any reasonable software,
367
00:24:15,120 --> 00:24:17,250
would compute a determinant.
368
00:24:17,250 --> 00:24:20,990
If I have a matrix
of size a hundred,
369
00:24:20,990 --> 00:24:24,817
the way I would actually
compute its determinant would be
370
00:24:24,817 --> 00:24:27,400
elimination, make it triangular,
multiply the pivots together,
371
00:24:27,400 --> 00:24:29,940
but it -- would
it be possible t-
372
00:24:29,940 --> 00:24:33,110
to produce the same matrix
the product of the pivots,
373
00:24:33,110 --> 00:24:34,080
the product of pivots.
374
00:24:34,080 --> 00:24:39,080
Now, there's always in
determinants a plus or minus
375
00:24:39,080 --> 00:24:44,630
and cross every T in that
proof, but that's really
376
00:24:44,630 --> 00:24:46,850
the sign to remember.
377
00:24:46,850 --> 00:24:51,840
Where -- where does that
come into this rule?
378
00:24:51,840 --> 00:24:54,500
Could it be, could
the determinant
379
00:24:54,500 --> 00:24:58,210
be minus the product
of the pivots?
380
00:24:58,210 --> 00:25:00,740
The determinant
is d1, d2, to dn.
381
00:25:00,740 --> 00:25:02,380
No doubt about that.
382
00:25:02,380 --> 00:25:05,370
But to get to this
triangular form,
383
00:25:05,370 --> 00:25:12,660
we may have had to do a row
exchange, so, so this --
384
00:25:12,660 --> 00:25:15,890
it's the product of the pivots
if there were no row exchanges.
385
00:25:15,890 --> 00:25:19,370
If, if SLU code,
the simple LU code,
386
00:25:19,370 --> 00:25:21,770
the square LU went
right through.
387
00:25:21,770 --> 00:25:24,070
If we had to do
some row exchanges,
388
00:25:24,070 --> 00:25:26,840
then we've got to
watch plus or minus.
389
00:25:26,840 --> 00:25:31,680
OK, but though -- this
law is simply that.
390
00:25:31,680 --> 00:25:33,180
OK, how do I prove that?
391
00:25:37,420 --> 00:25:42,730
Let's see, let me suppose
that d's are not zeroes.
392
00:25:42,730 --> 00:25:44,620
The pivots are not zeroes.
393
00:25:44,620 --> 00:25:50,550
And tell me, how do I show
that none of this upper stuff
394
00:25:50,550 --> 00:25:53,840
makes any difference?
395
00:25:53,840 --> 00:25:56,550
How do I get zeroes there?
396
00:25:56,550 --> 00:25:58,630
By elimination, right?
397
00:25:58,630 --> 00:26:01,940
I just multiply this
row by the right number,
398
00:26:01,940 --> 00:26:05,950
subtract from that
row, kills that.
399
00:26:05,950 --> 00:26:09,030
I multiply this row by the
right number, kills that,
400
00:26:09,030 --> 00:26:11,010
by the right number, kills that.
401
00:26:11,010 --> 00:26:15,850
Now, you kill every one of these
off-diagonal terms, no problem
402
00:26:15,850 --> 00:26:17,450
and I'm just left
with the diagonal.
403
00:26:20,690 --> 00:26:22,940
Well, strictly
speaking, I still have
404
00:26:22,940 --> 00:26:26,680
to figure out why is,
for a diagonal matrix
405
00:26:26,680 --> 00:26:28,745
now, why is that the
right determinant?
406
00:26:31,540 --> 00:26:37,260
I mean, we sure
hope it is, but why?
407
00:26:37,260 --> 00:26:41,270
I have to go back to
properties one, two, three.
408
00:26:41,270 --> 00:26:46,710
Why is -- now that the
matrix is suddenly diagonal,
409
00:26:46,710 --> 00:26:48,950
how do I know that the
determinant is just
410
00:26:48,950 --> 00:26:51,070
a product of That's
my proof, really,
411
00:26:51,070 --> 00:26:53,200
that once I've got
those diagonal entries?
412
00:26:53,200 --> 00:26:55,030
Well, what I going to use?
413
00:26:55,030 --> 00:26:57,760
I'm going to use property
3A, is that right?
414
00:26:57,760 --> 00:27:01,250
I'll factor this,
I'll factor this,
415
00:27:01,250 --> 00:27:05,380
I'll factor that d1 out
and have one and have
416
00:27:05,380 --> 00:27:07,280
the first row will be that.
417
00:27:07,280 --> 00:27:09,710
And then I'll factor out
the d2, shall I shall
418
00:27:09,710 --> 00:27:13,160
I put the d2 here,
and the second row
419
00:27:13,160 --> 00:27:16,110
will look like that, and so on.
420
00:27:16,110 --> 00:27:20,150
So I've factored out all the
d's and what I left with?
421
00:27:20,150 --> 00:27:21,370
The identity.
422
00:27:21,370 --> 00:27:24,970
And what rule do I
finally get to use?
423
00:27:24,970 --> 00:27:26,030
Rule one.
424
00:27:26,030 --> 00:27:29,900
Finally, this is the point
where rule one finally chips
425
00:27:29,900 --> 00:27:33,480
in and says that this
determinant is one,
426
00:27:33,480 --> 00:27:35,520
so it's the product of the d's.
427
00:27:35,520 --> 00:27:40,950
So this was rules five,
to do elimination,
428
00:27:40,950 --> 00:27:48,680
3A to factor out the D's, and,
and our best friend, rule one.
429
00:27:48,680 --> 00:27:52,240
And possibly rule
two, the exchanges
430
00:27:52,240 --> 00:27:53,710
may have been needed also.
431
00:27:53,710 --> 00:27:54,210
OK.
432
00:27:56,940 --> 00:28:01,350
Now I guess I have to consider
also the case if some d is
433
00:28:01,350 --> 00:28:06,680
zero, because I was assuming I
could use the d's to clean out
434
00:28:06,680 --> 00:28:08,490
and make a diagonal,
but what if --
435
00:28:08,490 --> 00:28:13,390
what if one of those
diagonal entries is zero?
436
00:28:13,390 --> 00:28:16,380
Well, then with
elimination we know
437
00:28:16,380 --> 00:28:21,440
that we can get a row
of zeroes, and for a row
438
00:28:21,440 --> 00:28:25,200
of zeroes I'm using rule
six, the determinant is zero,
439
00:28:25,200 --> 00:28:26,020
and that's right.
440
00:28:26,020 --> 00:28:28,690
So I can check
the singular case.
441
00:28:28,690 --> 00:28:36,250
In fact, I can now get to the
key point that determinant of A
442
00:28:36,250 --> 00:28:44,920
is zero, exactly when,
exactly when A is singular.
443
00:28:48,790 --> 00:28:52,880
And otherwise is not singular,
so that the determinant
444
00:28:52,880 --> 00:28:58,800
is a fair test for invertibility
or non-invertibility.
445
00:28:58,800 --> 00:29:03,610
So, I must be close to that
because I can take any matrix
446
00:29:03,610 --> 00:29:05,290
and get there.
447
00:29:05,290 --> 00:29:06,870
Do I -- did I have
anything to say?
448
00:29:09,570 --> 00:29:12,970
The, the proofs, it starts
by saying by elimination
449
00:29:12,970 --> 00:29:14,680
go from A to U.
450
00:29:14,680 --> 00:29:15,290
Oh, yeah.
451
00:29:15,290 --> 00:29:17,940
Actually looks to
me like I don't --
452
00:29:17,940 --> 00:29:22,450
haven't said anything brand-new
here, that, that really,
453
00:29:22,450 --> 00:29:28,950
I've got this, because
let's just remember the
454
00:29:28,950 --> 00:29:37,980
By elimination, I can go from
the original A to reason.
455
00:29:37,980 --> 00:29:43,440
Well, OK, now suppose the
matrix is U. singular.
456
00:29:43,440 --> 00:29:46,630
If the matrix is
singular, what happens?
457
00:29:46,630 --> 00:29:50,480
Then by elimination
I get a row of zeroes
458
00:29:50,480 --> 00:29:55,240
and therefore the
determinant is zero.
459
00:29:55,240 --> 00:29:59,170
And if the matrix is not
singular, I don't get zero,
460
00:29:59,170 --> 00:30:02,220
so maybe -- do you want me to
put this, like, in two parts?
461
00:30:02,220 --> 00:30:10,100
The determinant of A is not
zero when A is invertible.
462
00:30:15,480 --> 00:30:18,750
Because I've already --
463
00:30:18,750 --> 00:30:23,240
in chapter two we figured out
when is the matrix invertible.
464
00:30:23,240 --> 00:30:27,770
It's invertible when elimination
produces a full set of pivots
465
00:30:27,770 --> 00:30:31,420
and now, and we now, we know
the determinant is the product
466
00:30:31,420 --> 00:30:34,160
of those non-zero numbers.
467
00:30:34,160 --> 00:30:36,410
So those are the two cases.
468
00:30:36,410 --> 00:30:39,180
If it's singular, I
go to a row of zeroes.
469
00:30:43,370 --> 00:30:49,680
If it's invertible, I go to
U and then to the diagonal D,
470
00:30:49,680 --> 00:30:57,150
and then which -- and
then to d1, d2, up to dn.
471
00:30:57,150 --> 00:31:00,050
As the formula -- so
we have a formula now.
472
00:31:02,700 --> 00:31:05,110
We have a formula
for the determinant
473
00:31:05,110 --> 00:31:08,520
and it's actually a
very much more practical
474
00:31:08,520 --> 00:31:12,100
formula than the but they didn't
matter anyway. ad-bc formula.
475
00:31:12,100 --> 00:31:18,050
Is it correct, maybe you should
just -- let's just check that.
476
00:31:18,050 --> 00:31:18,970
Two-by-two.
477
00:31:18,970 --> 00:31:23,850
What are the pivots of
a two-by-two matrix?
478
00:31:23,850 --> 00:31:28,270
What does elimination do
with a two-by-two matrix?
479
00:31:28,270 --> 00:31:30,370
It -- there's the
first pivot, fine.
480
00:31:30,370 --> 00:31:33,880
What's the second pivot?
481
00:31:33,880 --> 00:31:38,970
We subtract, so I'm putting it
in this upper triangular form.
482
00:31:38,970 --> 00:31:44,270
What do I -- my multiplier
is c over a, right?
483
00:31:44,270 --> 00:31:46,790
I multiply that row
by c over a and I
484
00:31:46,790 --> 00:31:50,590
subtract to get that
zero, and here I
485
00:31:50,590 --> 00:31:54,470
have d minus c over a times b.
486
00:31:58,500 --> 00:32:01,870
That's the elimination
on a two-by-two.
487
00:32:01,870 --> 00:32:07,440
So I've finally discovered that
the determinant of this matrix
488
00:32:07,440 --> 00:32:07,940
--
489
00:32:07,940 --> 00:32:12,240
I've got it from the properties,
not by knowing the answer
490
00:32:12,240 --> 00:32:18,830
from last year, that the
determinant of this two-by-two
491
00:32:18,830 --> 00:32:22,730
is the product of A
times that, and of course
492
00:32:22,730 --> 00:32:26,180
when I multiply A by that,
the product of that and that
493
00:32:26,180 --> 00:32:30,485
is ad minus, the a is canceled,
494
00:32:30,485 --> 00:32:30,985
bc.
495
00:32:34,270 --> 00:32:36,320
So that's great,
provided a isn't zero.
496
00:32:36,320 --> 00:32:38,960
because all math professors
watching this will be waiting
497
00:32:38,960 --> 00:32:42,190
If a was zero, that step
wasn't allowed, with seven row
498
00:32:42,190 --> 00:32:45,130
exchanges and with ten row
exchanges? zero wasn't a pivot.
499
00:32:45,130 --> 00:32:46,600
So that's what I --
500
00:32:46,600 --> 00:32:48,850
I've covered all the bases.
501
00:32:48,850 --> 00:32:53,090
I have to -- if a is zero,
then I have to do the exchange,
502
00:32:53,090 --> 00:32:56,570
and if the exchange doesn't
work, it's because a is proof.
503
00:32:56,570 --> 00:32:57,730
singular.
504
00:32:57,730 --> 00:33:03,130
OK, those are --
505
00:33:03,130 --> 00:33:06,520
those are the direct
properties of the determinant.
506
00:33:06,520 --> 00:33:11,120
And now, finally, I've got
two more, nine and ten.
507
00:33:11,120 --> 00:33:13,920
And that's --
508
00:33:13,920 --> 00:33:15,030
I think you can...
509
00:33:15,030 --> 00:33:25,210
Like, the ones
we've got here are
510
00:33:25,210 --> 00:33:28,870
totally connected with
our elimination process
511
00:33:28,870 --> 00:33:35,340
and whether pivots are
available and whether we
512
00:33:35,340 --> 00:33:36,820
get a row of zeroes.
513
00:33:36,820 --> 00:33:40,360
I think all that you
can swallow in one shot.
514
00:33:40,360 --> 00:33:43,730
Let me tell you
properties nine and ten.
515
00:33:46,990 --> 00:33:50,000
They're quick to write down.
516
00:33:50,000 --> 00:33:53,940
They're very, very useful.
517
00:33:53,940 --> 00:33:56,260
So I'll just write
them down and use them.
518
00:33:56,260 --> 00:34:01,410
Property nine says that the
determinant of a product --
519
00:34:01,410 --> 00:34:05,610
if I That's the, like,
concrete proof that,
520
00:34:05,610 --> 00:34:07,180
multiply two matrices.
521
00:34:07,180 --> 00:34:11,909
So if I multiply two
matrices, A and B,
522
00:34:11,909 --> 00:34:14,050
that the determinant
of the product
523
00:34:14,050 --> 00:34:26,730
is determinant of A times
determinant of B, and for me
524
00:34:26,730 --> 00:34:31,750
that one is like, that's
a very valuable property,
525
00:34:31,750 --> 00:34:34,449
and it's sort of like partly
coming out of the blue,
526
00:34:34,449 --> 00:34:37,050
because we haven't been
multiplying matrices
527
00:34:37,050 --> 00:34:41,159
and here suddenly
this determinant
528
00:34:41,159 --> 00:34:44,590
has this multiplying property.
529
00:34:44,590 --> 00:34:46,750
Remember, it didn't have
the linear property,
530
00:34:46,750 --> 00:34:48,810
it didn't have the
adding property.
531
00:34:48,810 --> 00:34:52,389
The determinant
of A plus B is not
532
00:34:52,389 --> 00:34:57,210
the sum of the determinants,
but the determinant of A times B
533
00:34:57,210 --> 00:35:01,220
is the product, is the
product of the determinants.
534
00:35:01,220 --> 00:35:06,955
OK, so for example, what's
the determinant of A inverse?
535
00:35:12,560 --> 00:35:14,240
Using that property nine.
536
00:35:19,360 --> 00:35:21,230
Let me, let me put
that under here
537
00:35:21,230 --> 00:35:27,800
because the camera is happier
if it can focus on both at once.
538
00:35:27,800 --> 00:35:29,140
So let me put it underneath.
539
00:35:29,140 --> 00:35:34,510
The determinant of A
inverse, because property ten
540
00:35:34,510 --> 00:35:40,420
will come in that space.
541
00:35:40,420 --> 00:35:44,530
What does this tell me about
A inverse, its determinant?
542
00:35:44,530 --> 00:35:47,730
OK, well, what do I
know about A inverse?
543
00:35:47,730 --> 00:35:54,980
I know that A inverse
times A is odd.
544
00:35:54,980 --> 00:35:55,960
So what I going to do?
545
00:35:59,200 --> 00:36:02,450
I'm going to take
determinants of both sides.
546
00:36:02,450 --> 00:36:06,290
The determinant of
I is one, and what's
547
00:36:06,290 --> 00:36:10,390
the determinant of A inverse A?
548
00:36:10,390 --> 00:36:13,670
That's a product of
two matrices, A and B.
549
00:36:13,670 --> 00:36:15,510
So it's the product
of the determinant,
550
00:36:15,510 --> 00:36:16,830
so what I learning?
551
00:36:16,830 --> 00:36:18,840
I'm learning that
the determinant
552
00:36:18,840 --> 00:36:24,070
of A inverse times
the determinant of A
553
00:36:24,070 --> 00:36:27,800
is the determinant of
I, that's this one.
554
00:36:27,800 --> 00:36:31,870
Again, I happily
use property one.
555
00:36:31,870 --> 00:36:35,950
OK, so that tells me that
the determinant of A inverse
556
00:36:35,950 --> 00:36:37,270
is one over.
557
00:36:37,270 --> 00:36:40,000
Here's my -- here's
my conclusion --
558
00:36:40,000 --> 00:36:53,910
is one over the
determinant of A.
559
00:36:53,910 --> 00:36:55,810
I guess that that --
560
00:36:55,810 --> 00:36:59,620
I, I always try to think, well,
do we know some cases of that?
561
00:36:59,620 --> 00:37:04,450
Of course, we know it's right
already if A is diagonal.
562
00:37:04,450 --> 00:37:09,000
If A is a diagonal matrix,
then its determinant
563
00:37:09,000 --> 00:37:10,480
is just a product
of those numbers.
564
00:37:10,480 --> 00:37:14,890
So if A is, for
example, two-three,
565
00:37:14,890 --> 00:37:20,740
then we know that A-inverse
is one-half one-third,
566
00:37:20,740 --> 00:37:26,440
and sure enough, that
has determinant six,
567
00:37:26,440 --> 00:37:29,430
and that has
determinant one-sixth.
568
00:37:29,430 --> 00:37:32,360
And our rule checks.
569
00:37:32,360 --> 00:37:39,490
So somehow this proof,
this property has to --
570
00:37:39,490 --> 00:37:41,960
somehow the proof
of that property --
571
00:37:41,960 --> 00:37:45,950
if we can boil it down to
diagonal matrices then we can
572
00:37:45,950 --> 00:37:49,140
read it off, whether
it's A and A-inverse,
573
00:37:49,140 --> 00:37:52,660
or two different diagonal
matrices A and B.
574
00:37:52,660 --> 00:37:54,430
For diagonal --
so what I saying?
575
00:37:54,430 --> 00:37:59,050
I'm saying for a
diagonal matrices, check.
576
00:37:59,050 --> 00:38:02,380
But we'd have to do
elimination steps,
577
00:38:02,380 --> 00:38:08,510
we'd have to patiently
do the, the, argument
578
00:38:08,510 --> 00:38:11,810
if we want to use these
previous properties to get it
579
00:38:11,810 --> 00:38:12,980
for other matrices.
580
00:38:12,980 --> 00:38:18,094
And it also tells me -- what,
just let's, see what else
581
00:38:18,094 --> 00:38:18,760
it's telling me.
582
00:38:18,760 --> 00:38:21,900
What's the determinant
of, of A-squared?
583
00:38:21,900 --> 00:38:26,750
If I take a matrix
and square it?
584
00:38:26,750 --> 00:38:30,460
Then the determinant
just got squared.
585
00:38:30,460 --> 00:38:31,140
Right?
586
00:38:31,140 --> 00:38:34,180
The determinant of
A-squared is the determinant
587
00:38:34,180 --> 00:38:35,864
of A times the determinant of A.
588
00:38:35,864 --> 00:38:38,030
So if I square the matrix,
I square the determinant.
589
00:38:38,030 --> 00:38:43,180
If I double the matrix, what do
I do to the non-zeroes flipped
590
00:38:43,180 --> 00:38:46,350
to the other side of the
diagonal, determinant?
591
00:38:46,350 --> 00:38:47,930
Think about that one.
592
00:38:47,930 --> 00:38:52,690
If I double the matrix, what
-- so the determinant of A,
593
00:38:52,690 --> 00:38:56,150
since I'm writing down,
like, facts that follow,
594
00:38:56,150 --> 00:39:02,500
the determinant of A-squared
is the determinant of A,
595
00:39:02,500 --> 00:39:04,740
all squared.
596
00:39:04,740 --> 00:39:09,580
The determinant of 2A is what?
597
00:39:12,250 --> 00:39:16,380
That's A plus A.
598
00:39:16,380 --> 00:39:19,910
But wait, er, I
don't want the answer
599
00:39:19,910 --> 00:39:22,410
to determinant of A here.
600
00:39:22,410 --> 00:39:23,150
That's wrong.
601
00:39:23,150 --> 00:39:25,860
It's not two determinant
of A, What is it?
602
00:39:25,860 --> 00:39:28,880
OK, one more coming,
which I I have to make,
603
00:39:28,880 --> 00:39:32,190
what's the number that I have
to multiply determinant of A
604
00:39:32,190 --> 00:39:34,900
by if I double the
whole matrix, if I
605
00:39:34,900 --> 00:39:36,710
double every entry
in the matrix?
606
00:39:36,710 --> 00:39:38,220
What happens to the determinant?
607
00:39:38,220 --> 00:39:41,230
If that were possible,
that would be a bad thing,
608
00:39:41,230 --> 00:39:43,640
Supposed it's an n-by-n
matrix. that gets --
609
00:39:43,640 --> 00:39:44,850
get down to triangular
610
00:39:44,850 --> 00:39:46,690
Two to the n, right.
611
00:39:46,690 --> 00:39:48,120
Two to the nth.
612
00:39:48,120 --> 00:39:50,650
Now, why is it two to the
nth, and not just two?
613
00:39:54,700 --> 00:39:58,070
So why is it two to the nth?
614
00:39:58,070 --> 00:40:02,230
Because I'm factoring
out two from every row.
615
00:40:02,230 --> 00:40:05,610
There's a factor -- this has
a factor two in every row,
616
00:40:05,610 --> 00:40:08,640
so I can factor two
out of the first row.
617
00:40:08,640 --> 00:40:12,120
I factor a different two out of
the second row, a different two
618
00:40:12,120 --> 00:40:15,640
out of the nth row, so I've
got all those twos coming out.
619
00:40:15,640 --> 00:40:20,290
So it's like volume,
really, and that's
620
00:40:20,290 --> 00:40:23,570
one of the great
applications of determinants.
621
00:40:23,570 --> 00:40:30,860
If I -- if I have a box
and I double all the sides,
622
00:40:30,860 --> 00:40:35,800
I multiply the volume
by two to the nth.
623
00:40:35,800 --> 00:40:38,440
If it's a box in
three dimensions,
624
00:40:38,440 --> 00:40:40,970
I multiply the volume by 8.
625
00:40:43,710 --> 00:40:47,190
So this is rule 3A here.
626
00:40:47,190 --> 00:40:49,480
This is rule nine.
627
00:40:49,480 --> 00:40:55,100
And notice the way this
rule sort of checks out with
628
00:40:55,100 --> 00:41:02,550
the singular/non-singular
stuff, that if A is invertible,
629
00:41:02,550 --> 00:41:05,650
what does that mean
about its determinant?
630
00:41:05,650 --> 00:41:08,020
It's not zero, and
therefore this makes sense.
631
00:41:10,770 --> 00:41:12,940
The case when
determinant of A is
632
00:41:12,940 --> 00:41:19,870
zero, that's the case where my
formula doesn't work anymore.
633
00:41:19,870 --> 00:41:24,030
If determinant of A is
zero, this is ridiculous,
634
00:41:24,030 --> 00:41:25,800
and that's ridiculous.
635
00:41:25,800 --> 00:41:31,300
A-inverse doesn't exist, and one
over zero doesn't make sense.
636
00:41:31,300 --> 00:41:36,090
So don't miss this property.
637
00:41:36,090 --> 00:41:38,560
It's sort of, like,
amazing that it can...
638
00:41:38,560 --> 00:41:44,660
And the tenth property is
equally simple to state,
639
00:41:44,660 --> 00:41:47,720
that the determinant
of A transposed
640
00:41:47,720 --> 00:41:57,010
equals the determinant of A.
641
00:41:57,010 --> 00:42:03,030
And of course, let's just
check it on the ab cd guy.
642
00:42:03,030 --> 00:42:07,075
We could check that sure
enough, that's ab cd, it works.
643
00:42:09,710 --> 00:42:14,350
It's ad - bc, it's
ad - bc, sure enough.
644
00:42:14,350 --> 00:42:19,140
So that transposing did
not change the determinant.
645
00:42:19,140 --> 00:42:24,790
But what it does change is --
646
00:42:24,790 --> 00:42:28,400
well, what it does is
it lists, so all --
647
00:42:28,400 --> 00:42:31,340
I've been working with rows.
648
00:42:31,340 --> 00:42:35,690
If a row is all zeroes,
the determinant is zero.
649
00:42:35,690 --> 00:42:40,360
But now, with rule
ten, I know what to do
650
00:42:40,360 --> 00:42:42,750
is a column is all zero.
651
00:42:42,750 --> 00:42:46,550
If a column is all zero,
what's the determinant?
652
00:42:46,550 --> 00:42:48,260
Zero, again.
653
00:42:48,260 --> 00:42:53,250
So, like all those properties
about rows, exchanging two rows
654
00:42:53,250 --> 00:42:55,080
reverses the sign.
655
00:42:55,080 --> 00:42:58,210
Now, exchanging two
columns reverses
656
00:42:58,210 --> 00:43:00,990
the sign, because
I can always, if I
657
00:43:00,990 --> 00:43:03,690
want to see why,
I can transpose,
658
00:43:03,690 --> 00:43:08,680
those columns become rows, I do
the exchange, I transpose back.
659
00:43:08,680 --> 00:43:11,760
And I've done a
column operation.
660
00:43:11,760 --> 00:43:17,530
So, in, in conclusion, there was
nothing special about row one,
661
00:43:17,530 --> 00:43:20,610
'cause I could exchange
rows, and now there's
662
00:43:20,610 --> 00:43:25,060
nothing special about rows that
isn't equally true for columns
663
00:43:25,060 --> 00:43:26,980
because this is the same.
664
00:43:26,980 --> 00:43:27,580
OK.
665
00:43:27,580 --> 00:43:32,080
And again, maybe I won't --
666
00:43:32,080 --> 00:43:33,320
oh, let's see.
667
00:43:33,320 --> 00:43:33,820
Do we...?
668
00:43:33,820 --> 00:43:37,930
Maybe it's worth seeing a
quick proof of this number ten,
669
00:43:37,930 --> 00:43:44,620
quick, quick, er,
proof of number ten.
670
00:43:44,620 --> 00:43:48,970
Er, let me take the
-- this is number ten.
671
00:43:48,970 --> 00:43:51,380
A transposed equals A.
672
00:43:51,380 --> 00:43:56,480
Determinate of A transposed
equals determinate of A.
673
00:43:56,480 --> 00:43:58,110
That's what I should have said.
674
00:43:58,110 --> 00:43:59,280
OK.
675
00:43:59,280 --> 00:44:07,820
So, let's just, er.
676
00:44:07,820 --> 00:44:11,450
A typical matrix A,
if I use elimination,
677
00:44:11,450 --> 00:44:16,100
this factors into LU.
678
00:44:16,100 --> 00:44:21,710
And the transpose is U
transpose, l transpose.
679
00:44:25,430 --> 00:44:26,370
Er... let me.
680
00:44:29,150 --> 00:44:36,870
So this is proof, this is
proof number ten, using --
681
00:44:36,870 --> 00:44:39,820
well, I don't know which
ones I'll use, so I'll put
682
00:44:39,820 --> 00:44:42,840
'em all in, one to nine.
683
00:44:42,840 --> 00:44:43,650
OK.
684
00:44:43,650 --> 00:44:47,400
I'm going to prove number
ten by using one to nine.
685
00:44:47,400 --> 00:44:50,910
I won't cover every case,
but I'll cover almost every
686
00:44:50,910 --> 00:44:51,550
case.
687
00:44:51,550 --> 00:44:55,400
So in almost every case,
A can factor into LU,
688
00:44:55,400 --> 00:44:57,710
and A transposed can
factor into that.
689
00:44:57,710 --> 00:45:00,070
Now, what do I do next?
690
00:45:00,070 --> 00:45:03,910
So I want to prove that
these are the same.
691
00:45:03,910 --> 00:45:06,610
I see a product here.
692
00:45:06,610 --> 00:45:09,560
So I use rule nine.
693
00:45:09,560 --> 00:45:14,860
So, now what I want to prove is,
so now I know that this is LU,
694
00:45:14,860 --> 00:45:19,810
this is U transposed
and l transposed.
695
00:45:19,810 --> 00:45:24,010
Now, just for a practice, what
are all those determinants?
696
00:45:24,010 --> 00:45:28,710
So this is, this is, this is
prove this, prove this, prove
697
00:45:28,710 --> 00:45:32,460
this, and now I'm
ready to do it.
698
00:45:32,460 --> 00:45:34,460
What's the determinant of l?
699
00:45:34,460 --> 00:45:40,240
You remember what l is, it's
this lower triangular matrix
700
00:45:40,240 --> 00:45:43,085
with ones on the diagonals.
701
00:45:43,085 --> 00:45:44,710
So what is the
determinant of that guy?
702
00:45:44,710 --> 00:45:45,210
I- It's one.
703
00:45:48,640 --> 00:45:53,000
Any time I have this
triangular matrix,
704
00:45:53,000 --> 00:46:00,390
I can get rid of
all the non-zeroes,
705
00:46:00,390 --> 00:46:07,920
down to the diagonal case.
706
00:46:07,920 --> 00:46:12,750
The determinate of l is one.
707
00:46:12,750 --> 00:46:21,810
How about the determinant
of l transposed?
708
00:46:21,810 --> 00:46:25,410
That's triangular also, right?
709
00:46:25,410 --> 00:46:28,050
Still got those ones
on the diagonal,
710
00:46:28,050 --> 00:46:59,050
it's just the matrices and then
get down to diagonal matrices.
711
00:46:59,050 --> 00:47:00,600
right?
712
00:47:00,600 --> 00:47:09,290
If If I could --
why would it be bad?
713
00:47:09,290 --> 00:47:11,800
My whole lecture
would die, right?
714
00:47:11,800 --> 00:47:37,360
Because rule two said that if
you do seven row exchanges,
715
00:47:37,360 --> 00:47:47,700
then the sign of the
determinant reverses.
716
00:47:47,700 --> 00:47:56,820
But if you do ten row exchanges,
the sign of the determinant
717
00:47:56,820 --> 00:48:02,520
stays the same, because minus
one ten times is plus one.
718
00:48:02,520 --> 00:48:16,630
OK, so there's a hidden
fact here, that every --
719
00:48:16,630 --> 00:48:19,150
like, every permutation,
the permutations
720
00:48:19,150 --> 00:48:23,310
are either odd or even.
721
00:48:23,310 --> 00:48:26,360
I could get the permutation
with seven row exchanges,
722
00:48:26,360 --> 00:48:27,770
then I could
probably get it with
723
00:48:27,770 --> 00:48:31,430
twenty-one, or twenty-three,
or a hundred and one,
724
00:48:31,430 --> 00:48:33,430
if it's an odd one.
725
00:48:33,430 --> 00:48:35,980
Or an even number
of permutations, so,
726
00:48:35,980 --> 00:48:39,070
but that's the key
fact that just takes
727
00:48:39,070 --> 00:48:43,760
another little
algebraic trick to see,
728
00:48:43,760 --> 00:48:45,947
and that means that the
determinant is well-defined
729
00:48:45,947 --> 00:48:47,530
by properties one,
two, three and it's
730
00:48:47,530 --> 00:48:47,580
got properties four to ten.
731
00:48:47,580 --> 00:48:47,660
OK, that's today
and I'll try to get
732
00:48:47,660 --> 00:48:47,890
the homework for next Wednesday
onto the web this afternoon.
733
00:48:47,890 --> 00:48:49,440
Thanks.