WEBVTT
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OK, this is the second
lecture on determinants.
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There are only three.
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With determinants it's a
fascinating, small topic
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inside linear algebra.
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Used to be determinants
were the big thing,
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and linear algebra was the
little thing, but they --
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those changed, that
situation changed.
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Now determinants is one specific
part, very neat little part.
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And my goal today is to find
a formula for the determinant.
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It'll be a messy formula.
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So that's why you didn't
see it right away.
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But if I'm given
this n by n matrix
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then I use those
entries to create
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this number, the determinant.
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So there's a formula for it.
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In fact, there's another
formula, a second formula using
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something called cofactors.
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So you'll -- you have to
know what cofactors are.
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And then I'll apply
those formulas
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for some, some matrices
that have a lot of zeros
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away from the three diagonals.
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OK.
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So I'm shooting now for a
formula for the determinant.
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You remember we started with
these three properties, three
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simple properties,
but out of that we
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got all these amazing facts,
like the determinant of A B
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equals determinant of A
times determinant of B.
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But the three facts were --
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oh, how about I just
take two by twos.
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I know, because everybody here
knows, the determinant of a two
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by two matrix, but let's get
it out of these three formulas.
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OK, so here's my, my
two by two matrix.
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I'm looking for a formula
for this determinant.
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a b c d, OK.
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So property one, I know what
to do with the identity.
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Right?
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Property two allows
me to exchange rows,
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and I know what to do then.
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So I know that that
determinant is one.
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Property two allows me
to exchange rows and know
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that this determinant
is minus one.
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And now I want to use property
three to get everybody,
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to get everybody.
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And how will I do that?
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OK.
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So remember that if I keep
the second row the same,
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I'm allowed to use
linearity in the first row.
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And I'll just use
it in a simple way.
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I'll write this vector
a b as a 0 + 0 b.
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So that's one step using
property three, linearity
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in the first row when the
second row's the same.
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OK.
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But now you can guess
what I'm going to do next.
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I'll -- because I'd like to --
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if I can make the
matrices diagonal,
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then I'm clearly there.
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So I'll take this one.
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Now I'll keep the first row
fixed and split the second row,
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so that'll be an a 0
and I'll split that
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into a c 0 and, keeping that
first row the same, a 0 d.
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I used, for this
part, linearity.
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And now I'll -- whoops, that's
plus because I've got more
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coming.
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This one I'll do the same.
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I'll keep this
first row the same
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and I'll split c d
into c 0 and 0 d.
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OK.
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Now I've got four
easy determinants,
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and two of them are --
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well, all four are
extremely easy.
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Two of them are so easy as
to turn into zero, right?
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Which two of these determinants
are zero right away?
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The first guy is zero.
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Why is he zero?
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Why is that determinant
nothing, forget him?
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Well, it has a column of zeros.
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And by the -- well, so
one way to think is, well,
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it's a singular matrix.
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Oh, for, for like forty-eight
different reasons,
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that determinant is zero.
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It's a singular matrix
that has a column of zeros.
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It's, it's dead.
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And this one is
about as dead too.
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Column of zeros.
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OK.
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So that's leaving
us with this one.
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Now what do I -- how do
I know its determinant,
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following the rules?
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Well, I guess one of the
properties that we actually got
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to was the determinant of that
-- diagonal matrix, then --
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so I, I'm finally getting to
that determinant is the a d.
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And this determinant is what?
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What's this one?
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Minus -- because I would use
property two to do a flip
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to make it c b, then property
three to factor out the b,
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property c to
factor out the c --
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the property again to factor
out the c, and that minus,
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and of course finally I got the
answer that we knew we would
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get.
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But you see the method.
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You see the method, because it's
method I'm looking for here,
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not just a two by two answer
but the method of doing --
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now I can do three by threes
and four by fours and any size.
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So if you can see the method
of taking each row at a time --
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so let's -- what would
happen with three by threes?
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Can we mentally do
it rather than I
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write everything on the
board for three by threes?
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So what would we do if
I had three by threes?
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I would keep rows two
and three the same
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and I would split the first
row into how many pieces?
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Three pieces.
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I'd have an A zero
zero and a zero B zero
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and a zero zero C or
something for the first row.
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So I would instead of going from
one piece to two pieces to four
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pieces, I would go from one
piece to three pieces to --
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what would it be?
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Each of those three,
would, would it be nine?
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Or twenty-seven?
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Oh yeah, I've actually
got more steps,
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right.
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I'd go to nine but then I'd have
another row to straighten out,
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twenty-seven.
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Yes, oh God.
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OK, let me say this again then.
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If I -- if it was three
by three, I would --
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separating out one
row into three pieces
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would give me three, separating
out the second row into three
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pieces, then I'd be up to nine,
separating out the third row
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into its three pieces, I'd be
up to twenty-seven, three cubed,
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pieces.
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But a lot of them would be zero.
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So now when would
they not be zero?
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Tell me the pieces
that would not be zero.
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Now I will write
the non-zero ones.
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OK, so I have this matrix.
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I think I have to
use these, start
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using these double symbols here
because otherwise I could never
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do n by n.
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OK.
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OK.
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So I split this up like crazy.
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A bunch of pieces are zero.
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Whenever I have a column of
zeros, I know I've got zero.
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When do I not have zero?
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When do I have -- what is
it that's like these guys?
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These are the survivors,
two survivors there.
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So my question
for three by three
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is going to be what
are the survivors?
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How many survivors are there?
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What are they?
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And when do I get a survivor.
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Well, I would get a survivor --
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for example, one
survivor will be
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that one times that one times
that one, with all zeros
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everywhere else.
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That would be one survivor.
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a one one zero zero
zero a two two zero zero
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zero a three three.
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That's like the a d survivor.
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Tell me another survivor.
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What other thing -- oh,
now here you see the clue.
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Now can -- shall I just
say the whole clue?
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That I'm having --
00:09:02.920 --> 00:09:10.460
the survivors have one entry
from each row and each column.
00:09:10.460 --> 00:09:14.430
One entry from each
row and column.
00:09:14.430 --> 00:09:16.760
Because if some
column is missing,
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then I get a singular matrix.
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And that, that's
one of these guys.
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See, you see what
happened with --
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this guy?
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Column one never
got used in 0 b 0 d.
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So its determinant was
zero and I forget it.
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So I'm going to forget
those and just put --
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so tell me one more that
would be a survivor?
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Well -- well,
here's another one.
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a one one zero zero -- now OK,
that's used up row -- row one
00:09:50.690 --> 00:09:51.800
is used.
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Column one is already
used so it better be zero.
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What else could I have?
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Where could I pick the guy --
which column shall I use in row
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two?
00:10:04.360 --> 00:10:08.070
Use column three, because
here if I use column --
00:10:08.070 --> 00:10:10.140
here I used column
one and row one.
00:10:10.140 --> 00:10:12.710
This was like the column --
00:10:12.710 --> 00:10:15.950
numbers were one two
three, right in order.
00:10:15.950 --> 00:10:23.140
Now the column numbers are going
to be one three, column three,
00:10:23.140 --> 00:10:26.070
and column two.
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So the row numbers are
one two three, of course.
00:10:29.620 --> 00:10:32.100
The column numbers are some --
00:10:32.100 --> 00:10:35.550
OK, some permutation
of one two three,
00:10:35.550 --> 00:10:38.880
and here they come in
the order one three two.
00:10:38.880 --> 00:10:42.060
It's just like having
a permutation matrix
00:10:42.060 --> 00:10:46.330
with, instead of the
ones, with numbers.
00:10:46.330 --> 00:10:50.940
And actually, it's very close
to having a permutation matrix,
00:10:50.940 --> 00:10:55.550
because I, what I do eventually
is I factor out these numbers
00:10:55.550 --> 00:10:57.630
and then I have got.
00:10:57.630 --> 00:10:59.520
So what is that
determinant equal?
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I factor those
numbers out and I've
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got a one one times a two
two times a three three.
00:11:05.400 --> 00:11:08.020
And what does this
determinant equal?
00:11:08.020 --> 00:11:09.530
Yeah, now tell me the, this --
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I mean, we're really getting
to the heart of these formulas
00:11:13.140 --> 00:11:13.690
now.
00:11:13.690 --> 00:11:16.800
What is that determinant?
00:11:16.800 --> 00:11:20.400
By the laws of -- by,
by our three properties,
00:11:20.400 --> 00:11:24.920
I can factor these out, I
can factor out the a one one,
00:11:24.920 --> 00:11:27.780
the a two three,
and the a three two.
00:11:27.780 --> 00:11:28.980
They're in separate rows.
00:11:28.980 --> 00:11:31.960
I can do each row separately.
00:11:31.960 --> 00:11:35.230
And then I just have to
decide is that plus sign
00:11:35.230 --> 00:11:37.530
or is that a minus sign?
00:11:37.530 --> 00:11:42.640
And the answer is it's a minus.
00:11:42.640 --> 00:11:43.490
Why minus?
00:11:43.490 --> 00:11:47.350
Because these is
one row exchange
00:11:47.350 --> 00:11:49.780
to get it back to the identity.
00:11:49.780 --> 00:11:52.740
So that's a minus.
00:11:52.740 --> 00:11:53.960
Now I through?
00:11:53.960 --> 00:11:55.455
No, because there
are other ways.
00:11:59.020 --> 00:12:02.060
What I'm really
through with, what
00:12:02.060 --> 00:12:04.510
I've done, what I've,
what I've completed
00:12:04.510 --> 00:12:08.110
is only the part where
the a one one is there.
00:12:08.110 --> 00:12:11.810
But now I've got parts
where it's a one two.
00:12:15.010 --> 00:12:18.820
And now if it's a one two that
row is used, that column is
00:12:18.820 --> 00:12:19.610
used.
00:12:19.610 --> 00:12:21.260
You see that idea?
00:12:21.260 --> 00:12:25.380
I could use this row and column.
00:12:25.380 --> 00:12:28.070
Now that column is used,
that column is used,
00:12:28.070 --> 00:12:31.300
and this guy has to be
here, a three three.
00:12:31.300 --> 00:12:33.190
And what's that determinant?
00:12:33.190 --> 00:12:38.510
That's an a one two times an a
two one times an a three three,
00:12:38.510 --> 00:12:42.510
and does it have
a plus or a minus?
00:12:42.510 --> 00:12:43.820
A minus is right.
00:12:43.820 --> 00:12:45.600
It has a minus.
00:12:45.600 --> 00:12:47.690
Because it's one
flip away from an id-
00:12:47.690 --> 00:12:51.730
from the, regular, the right
order, the diagonal order.
00:12:51.730 --> 00:12:54.200
And now what's the other
guy with a -- with,
00:12:54.200 --> 00:12:57.470
a one two up there?
00:12:57.470 --> 00:12:59.240
I could have used this row.
00:12:59.240 --> 00:13:05.080
I could have put this guy
here and this guy here.
00:13:05.080 --> 00:13:05.580
Right?
00:13:05.580 --> 00:13:07.730
You see the whole deal?
00:13:07.730 --> 00:13:14.030
Now that's an a one two,
a two three, a three one,
00:13:14.030 --> 00:13:17.970
and does that go with
a plus or a minus?
00:13:17.970 --> 00:13:19.720
Yeah, now that takes
a minute of thinking,
00:13:19.720 --> 00:13:23.210
doesn't it, because one
row exchange doesn't get it
00:13:23.210 --> 00:13:24.760
in line.
00:13:24.760 --> 00:13:26.300
So what is the answer for this?
00:13:26.300 --> 00:13:28.090
Plus or minus?
00:13:28.090 --> 00:13:32.120
Plus, because it
takes two exchanges.
00:13:32.120 --> 00:13:35.920
I could exchange rows one and
three and then two and three.
00:13:35.920 --> 00:13:40.360
Two exchanges makes
this thing a plus.
00:13:40.360 --> 00:13:43.300
And then finally we have --
we're going to have two more.
00:13:43.300 --> 00:13:43.800
OK.
00:13:43.800 --> 00:13:52.770
Zero zero a one three, a two
one zero zero, zero a three two
00:13:52.770 --> 00:13:54.160
zero.
00:13:54.160 --> 00:13:55.720
And one more guy.
00:13:55.720 --> 00:14:00.820
Zero zero a one
three, zero a two
00:14:00.820 --> 00:14:06.750
two zero, A three one zero zero.
00:14:06.750 --> 00:14:08.870
And let's put down
what we get from those.
00:14:08.870 --> 00:14:14.160
An a one three, an a two one,
and an a three two, and I
00:14:14.160 --> 00:14:16.650
think that one is a plus.
00:14:16.650 --> 00:14:21.510
And this guys is a minus because
one exchange would put it --
00:14:21.510 --> 00:14:24.864
would order it.
00:14:24.864 --> 00:14:25.655
And that's a minus.
00:14:29.230 --> 00:14:32.510
All right, that has
taken one whole board
00:14:32.510 --> 00:14:35.030
just to do the three by three.
00:14:35.030 --> 00:14:37.630
But do you agree
that we now have
00:14:37.630 --> 00:14:42.590
a formula for the
determinant which
00:14:42.590 --> 00:14:44.080
came from the three properties?
00:14:46.840 --> 00:14:50.440
And it must be it.
00:14:50.440 --> 00:14:53.090
And I'm going to
keep that formula.
00:14:53.090 --> 00:14:57.870
That's a famous -- that three
by three formula is one that
00:14:57.870 --> 00:15:01.570
if, if the cameras will follow
me back to the beginning here,
00:15:01.570 --> 00:15:07.130
I, I get the ones with the plus
sign are the ones that go down
00:15:07.130 --> 00:15:08.800
like down this way.
00:15:08.800 --> 00:15:10.390
And the ones with
the minus signs
00:15:10.390 --> 00:15:13.940
are sort of the ones
that go this way.
00:15:13.940 --> 00:15:17.440
I won't make that precise.
00:15:17.440 --> 00:15:20.700
For two reasons, one,
it would clutter up
00:15:20.700 --> 00:15:24.970
the board, and second reason,
it wouldn't be right for four
00:15:24.970 --> 00:15:26.300
by fours.
00:15:26.300 --> 00:15:29.540
For four by four, let
me just say right away,
00:15:29.540 --> 00:15:33.360
four by four matrix --
the, the cross diagonal,
00:15:33.360 --> 00:15:38.190
the wrong diagonal happens
to come out with a plus sign.
00:15:38.190 --> 00:15:39.800
Why is that?
00:15:39.800 --> 00:15:43.600
If I have a four by
four matrix with ones
00:15:43.600 --> 00:15:50.080
coming on the counter diagonal,
that determinant is plus.
00:15:50.080 --> 00:15:50.980
Why?
00:15:50.980 --> 00:15:54.980
Why plus for that guy?
00:15:54.980 --> 00:15:59.720
Because if I exchange
rows one and four and then
00:15:59.720 --> 00:16:02.850
I exchange rows two and
three, I've got the identity,
00:16:02.850 --> 00:16:05.590
and I did two exchanges.
00:16:05.590 --> 00:16:10.880
So this down to this, like,
you know, down toward Miami
00:16:10.880 --> 00:16:16.330
and down toward LA stuff is,
like, three by three only.
00:16:16.330 --> 00:16:16.830
OK.
00:16:19.510 --> 00:16:25.185
But I do want to get now --
00:16:25.185 --> 00:16:27.310
I don't want to go through
this for a four by four.
00:16:29.990 --> 00:16:34.280
I do want to get now
the general formula.
00:16:34.280 --> 00:16:39.570
So this is what I refer to in
the book as the big formula.
00:16:39.570 --> 00:16:43.140
So now this is the big
formula for the determinant.
00:16:43.140 --> 00:16:47.380
I'm asking you to make a jump
from two by two and three
00:16:47.380 --> 00:16:50.260
by three to n by n.
00:16:50.260 --> 00:16:52.220
OK, so this will
be the big formula.
00:17:00.250 --> 00:17:07.310
That the determinant of A is
the sum of a whole lot of terms.
00:17:07.310 --> 00:17:09.550
And what are those terms?
00:17:09.550 --> 00:17:12.490
And, and is it a
plus or a minus sign,
00:17:12.490 --> 00:17:14.869
and I have to tell you
which, which it is,
00:17:14.869 --> 00:17:18.520
because this came in -- in
the three by three case,
00:17:18.520 --> 00:17:20.140
I had how many terms?
00:17:20.140 --> 00:17:21.859
Six.
00:17:21.859 --> 00:17:25.630
And half were plus
and half were minus.
00:17:25.630 --> 00:17:30.570
How many terms are you
figuring for four by four?
00:17:30.570 --> 00:17:36.520
If I get two terms in the
two by two case, three --
00:17:36.520 --> 00:17:41.760
six terms in the three by three
case, what's that pattern?
00:17:41.760 --> 00:17:44.210
How many terms in the
four by four case?
00:17:46.830 --> 00:17:48.320
Twenty-four.
00:17:48.320 --> 00:17:49.715
Four factorial.
00:17:52.430 --> 00:17:53.750
Why four factorial?
00:17:53.750 --> 00:17:56.200
This will be a sum
of n factorial terms.
00:18:00.000 --> 00:18:01.530
Twenty-four, a
hundred and twenty,
00:18:01.530 --> 00:18:05.570
seven hundred and twenty,
whatever's after that.
00:18:05.570 --> 00:18:06.580
OK.
00:18:06.580 --> 00:18:08.945
Half plus and half minus.
00:18:12.110 --> 00:18:14.720
And where do those n
factorial -- terms come from?
00:18:14.720 --> 00:18:17.310
This is the moment to
listen to this lecture.
00:18:17.310 --> 00:18:20.110
Where do those n
factorial terms come from?
00:18:20.110 --> 00:18:23.690
They come because the first,
the guy in the first row
00:18:23.690 --> 00:18:26.940
can be chosen n ways.
00:18:26.940 --> 00:18:33.390
And after he's chosen, that's
used up that, that column.
00:18:33.390 --> 00:18:38.440
So the one in the second row
can be chosen n minus one ways.
00:18:38.440 --> 00:18:42.360
And after she's chosen,
that second column has been
00:18:42.360 --> 00:18:43.160
used.
00:18:43.160 --> 00:18:46.650
And then the one in the third
row can be chosen n minus two
00:18:46.650 --> 00:18:49.140
ways, and after it's chosen --
00:18:49.140 --> 00:18:52.330
notice how I'm getting
these personal pronouns.
00:18:52.330 --> 00:18:53.510
But I've run out.
00:18:53.510 --> 00:18:59.570
And I'm not willing to
stop with three by three,
00:18:59.570 --> 00:19:02.370
so I'm just going to
write the formula down.
00:19:02.370 --> 00:19:07.470
So the one in the first row
comes from some column alpha.
00:19:07.470 --> 00:19:10.670
I don't know what alpha is.
00:19:10.670 --> 00:19:11.600
And the one in the --
00:19:11.600 --> 00:19:14.410
I multiply that by somebody
in the second row that comes
00:19:14.410 --> 00:19:16.530
from some different column.
00:19:16.530 --> 00:19:19.470
And I multiply that by somebody
in the third row who comes
00:19:19.470 --> 00:19:21.850
from some yet different column.
00:19:21.850 --> 00:19:25.269
And then in the n-th
row, I don't know what --
00:19:25.269 --> 00:19:26.310
I don't know how to draw.
00:19:26.310 --> 00:19:29.570
Maybe omega, for last.
00:19:29.570 --> 00:19:32.170
And the whole point
is then that --
00:19:32.170 --> 00:19:34.510
that those column
numbers are different,
00:19:34.510 --> 00:19:40.990
that alpha, beta, gamma, omega,
that set of column numbers
00:19:40.990 --> 00:19:50.100
is some permutation,
permutation of one to n.
00:19:50.100 --> 00:19:54.830
It, it, the n column
numbers are each used once.
00:19:54.830 --> 00:19:57.570
And that gives us
n factorial terms.
00:19:57.570 --> 00:20:02.130
And when I choose
a term, that means
00:20:02.130 --> 00:20:04.850
I'm choosing somebody
from every row and column.
00:20:04.850 --> 00:20:10.300
And then I just -- like the way
I had this from row and column
00:20:10.300 --> 00:20:14.160
one, row and column two, row
and column three, so that --
00:20:14.160 --> 00:20:19.080
what was the alpha beta stuff
in that, for that term here?
00:20:19.080 --> 00:20:22.360
Alpha was one, beta was
two, gamma was three.
00:20:22.360 --> 00:20:26.180
The permutation was, was
the trivial permutation, one
00:20:26.180 --> 00:20:28.140
two three, everybody
in the right order.
00:20:30.730 --> 00:20:31.750
You see that formula?
00:20:34.420 --> 00:20:37.120
It's -- do you see why I
didn't want to start with that
00:20:37.120 --> 00:20:39.930
the first day, Friday?
00:20:39.930 --> 00:20:42.470
I'd rather we understood
the properties.
00:20:42.470 --> 00:20:44.920
Because out of this
formula, presumably I
00:20:44.920 --> 00:20:47.720
could figure out all
these properties.
00:20:47.720 --> 00:20:50.900
How would I know that the
determinant of the identity
00:20:50.900 --> 00:20:56.630
matrix was one, for example,
out of this formula?
00:20:56.630 --> 00:20:59.680
Why is -- if A is
the identity matrix,
00:20:59.680 --> 00:21:04.080
how does this formula
give me a plus one?
00:21:04.080 --> 00:21:05.160
You see it, right?
00:21:05.160 --> 00:21:10.440
Because, because almost
all the terms are zeros.
00:21:10.440 --> 00:21:15.450
Which term isn't zero, if,
if A is the identity matrix?
00:21:15.450 --> 00:21:18.501
Almost all the terms are zero
because almost all the As are
00:21:18.501 --> 00:21:19.000
zero.
00:21:19.000 --> 00:21:21.070
It's only, the only
time I'll get something
00:21:21.070 --> 00:21:25.270
is if it's a one one times a
two two times a three three.
00:21:25.270 --> 00:21:28.590
Only, only the,
only the permutation
00:21:28.590 --> 00:21:32.650
that's in the right order
will, will give me something.
00:21:32.650 --> 00:21:34.400
It'll come with a plus sign.
00:21:34.400 --> 00:21:37.490
And the determinant of
the identity is one.
00:21:37.490 --> 00:21:40.910
So, so we could go back
from this formula and prove
00:21:40.910 --> 00:21:41.710
everything.
00:21:41.710 --> 00:21:45.460
We could even try to prove
that the determinant of A B
00:21:45.460 --> 00:21:48.860
was the determinant of A
times the determinant of B.
00:21:48.860 --> 00:21:51.070
But like next week we would
still be working on it,
00:21:51.070 --> 00:21:54.600
because it's not --
00:21:54.600 --> 00:21:56.500
clear from -- if I took A B,
00:21:56.500 --> 00:21:57.030
my God.
00:21:57.030 --> 00:21:57.530
You know --.
00:21:57.530 --> 00:22:02.210
The entries of A B would
be all these pieces.
00:22:02.210 --> 00:22:06.630
Well, probably, it's probably
-- historically it's been done,
00:22:06.630 --> 00:22:09.131
but it won't be repeated
in eighteen oh six.
00:22:09.131 --> 00:22:09.630
OK.
00:22:09.630 --> 00:22:16.480
It would be possible probably
to see, why the determinant of A
00:22:16.480 --> 00:22:18.190
equals the determinant
of A transpose.
00:22:18.190 --> 00:22:21.154
That was another, like,
miracle property at the end.
00:22:21.154 --> 00:22:22.820
That would, that
would, that's an easier
00:22:22.820 --> 00:22:25.290
one, which we could find.
00:22:25.290 --> 00:22:26.200
OK.
00:22:26.200 --> 00:22:30.460
Is that all right
for the big formula?
00:22:30.460 --> 00:22:33.160
I could take you
then a, a typical --
00:22:33.160 --> 00:22:36.070
let me do an example.
00:22:36.070 --> 00:22:39.280
Which I'll just create.
00:22:39.280 --> 00:22:42.410
I'll take a four by four matrix.
00:22:42.410 --> 00:22:46.820
I'll put some, I'll put some
ones in and some zeros in.
00:22:46.820 --> 00:22:47.350
OK.
00:22:47.350 --> 00:22:48.530
Let me --
00:22:48.530 --> 00:22:52.690
I don't know how many to
put in, to tell the truth.
00:22:52.690 --> 00:22:54.100
I've never done this before.
00:22:58.840 --> 00:23:02.260
I don't know the
determinant of that matrix.
00:23:02.260 --> 00:23:05.380
So like mathematics is being
done for the first time
00:23:05.380 --> 00:23:07.772
in, in front of your eyes.
00:23:07.772 --> 00:23:08.730
What's the determinant?
00:23:12.090 --> 00:23:15.210
Well, a lot of -- there
are twenty-four terms,
00:23:15.210 --> 00:23:17.700
because it's four by four.
00:23:17.700 --> 00:23:19.460
Many of them will be
zero, because I've
00:23:19.460 --> 00:23:22.430
got all those zeros there.
00:23:22.430 --> 00:23:25.350
Maybe the whole
determinant is zero.
00:23:25.350 --> 00:23:29.390
I mean, I -- is that
a singular matrix?
00:23:29.390 --> 00:23:33.470
That possibility
definitely exists.
00:23:33.470 --> 00:23:37.620
I could, I could, So one way
to do it would be elimination.
00:23:37.620 --> 00:23:42.090
Actually, that would probably
be a fairly reasonable way.
00:23:42.090 --> 00:23:44.440
I could use elimination,
so I could use --
00:23:44.440 --> 00:23:47.760
go back to those properties,
that -- and use elimination,
00:23:47.760 --> 00:23:50.990
get down, eliminate it down,
do I have a row of zeros
00:23:50.990 --> 00:23:53.020
at the end of elimination?
00:23:53.020 --> 00:23:54.230
The answer is zero.
00:23:54.230 --> 00:23:58.320
I was thinking, shall
I try this big formula?
00:23:58.320 --> 00:23:59.390
OK.
00:23:59.390 --> 00:24:00.810
Let's try the big formula.
00:24:00.810 --> 00:24:08.190
How -- tell me one way I can go
down the matrix, taking a one,
00:24:08.190 --> 00:24:13.250
taking a one from every row and
column, and make it to the end?
00:24:13.250 --> 00:24:15.640
So it's -- I get
something that isn't zero.
00:24:15.640 --> 00:24:18.280
Well, one way to do it, I could
take that times that times
00:24:18.280 --> 00:24:20.360
that times that times that.
00:24:20.360 --> 00:24:23.510
That would be one and,
and, and I just said,
00:24:23.510 --> 00:24:25.990
that comes in with what sign?
00:24:25.990 --> 00:24:26.820
Plus.
00:24:26.820 --> 00:24:28.690
That comes with a plus sign.
00:24:28.690 --> 00:24:32.600
Because, because
that permutation --
00:24:32.600 --> 00:24:35.160
I've just written
the permutation
00:24:35.160 --> 00:24:38.530
about four three two
one, and one exchange
00:24:38.530 --> 00:24:40.900
and a second exchange,
two exchanges
00:24:40.900 --> 00:24:42.540
puts it in the correct order.
00:24:46.750 --> 00:24:51.620
Keep walking away, don't....
00:24:51.620 --> 00:24:54.550
OK, we're executing a
determinant formula here.
00:24:54.550 --> 00:25:07.900
Uh as long as it's not
periodic, of course.
00:25:07.900 --> 00:25:11.290
If he comes back I'm in --
00:25:11.290 --> 00:25:11.790
no.
00:25:11.790 --> 00:25:13.620
All right, all right.
00:25:13.620 --> 00:25:16.260
OK, so that would
give me a plus one.
00:25:21.260 --> 00:25:22.700
All right.
00:25:22.700 --> 00:25:24.240
Are there any others?
00:25:24.240 --> 00:25:26.610
Well, of course we
see another one here.
00:25:26.610 --> 00:25:29.840
This times this times this
times this strikes us right
00:25:29.840 --> 00:25:30.340
away.
00:25:30.340 --> 00:25:34.470
So that's the order
three, the order --
00:25:34.470 --> 00:25:37.450
let me make a little
different mark here.
00:25:37.450 --> 00:25:41.520
Three two one four.
00:25:41.520 --> 00:25:45.330
And is that a plus or a
minus, three two one four?
00:25:48.030 --> 00:25:52.950
Is that, is that permutation
a plus or a minus permutation?
00:25:52.950 --> 00:25:53.827
It's a minus.
00:25:53.827 --> 00:25:54.660
How do you see that?
00:25:54.660 --> 00:25:59.640
What exchange shall I do to
get it in the right order?
00:25:59.640 --> 00:26:02.480
If I exchange the one and the
three I'm in the right orders,
00:26:02.480 --> 00:26:05.120
took one exchange to do it,
so that would be a plus --
00:26:05.120 --> 00:26:07.040
that would be a minus one.
00:26:07.040 --> 00:26:09.700
And now I don't know if
there're any more here.
00:26:09.700 --> 00:26:10.280
Let's see.
00:26:10.280 --> 00:26:15.670
Let me try again
starting with this.
00:26:15.670 --> 00:26:18.360
Now I've got to pick somebody
from -- oh yeah, see,
00:26:18.360 --> 00:26:20.400
you see what's happening.
00:26:20.400 --> 00:26:24.900
If I I start there, OK,
column three is used.
00:26:24.900 --> 00:26:27.680
So then when I go to next
row, I can't use that,
00:26:27.680 --> 00:26:28.850
I must use that.
00:26:28.850 --> 00:26:30.810
Now columns two
and three are used.
00:26:30.810 --> 00:26:33.370
When I come to this
row I must use that.
00:26:33.370 --> 00:26:34.700
And then I must use that.
00:26:34.700 --> 00:26:38.280
So if I start there, this
is the only one I get.
00:26:38.280 --> 00:26:42.220
And similarly, if I start there,
that's the only one I get.
00:26:42.220 --> 00:26:45.280
So what's the determinant?
00:26:45.280 --> 00:26:47.420
What's the determinant?
00:26:47.420 --> 00:26:47.920
Zero.
00:26:47.920 --> 00:26:51.940
The determinant is
zero for that case.
00:26:51.940 --> 00:26:56.080
Because we, we were able to
check the twenty-four terms.
00:26:56.080 --> 00:26:57.850
Twenty-two of them were zero.
00:26:57.850 --> 00:26:59.290
One of them was plus one.
00:26:59.290 --> 00:27:01.100
One of them was minus one.
00:27:01.100 --> 00:27:04.600
Add up the twenty-four
terms, zero is the answer.
00:27:04.600 --> 00:27:05.120
OK.
00:27:05.120 --> 00:27:07.136
Well, I didn't know
it would be zero, I --
00:27:07.136 --> 00:27:08.760
because I wasn't,
like, thinking ahead.
00:27:08.760 --> 00:27:10.700
I was a little scared, actually.
00:27:13.300 --> 00:27:18.360
I said, that,
apparition went by.
00:27:18.360 --> 00:27:22.420
So and I don't know if
the camera caught that.
00:27:22.420 --> 00:27:23.970
So whether the rest
of the world will
00:27:23.970 --> 00:27:27.450
realize that I was in danger
or not, we don't know.
00:27:27.450 --> 00:27:29.674
But anyway, I guess
he just wanted
00:27:29.674 --> 00:27:31.590
to be sure that we got
the right answer, which
00:27:31.590 --> 00:27:33.260
is determinant zero.
00:27:33.260 --> 00:27:36.060
And then that makes me
think, OK, the matrix
00:27:36.060 --> 00:27:41.110
must be, the matrix
must be singular.
00:27:41.110 --> 00:27:43.020
And then if the
matrix is singular,
00:27:43.020 --> 00:27:46.000
maybe there's another way to see
that it's singular, like find
00:27:46.000 --> 00:27:47.225
something in its null space.
00:27:50.690 --> 00:27:54.190
Or find a combination of
the rows that gives zero.
00:27:54.190 --> 00:27:58.320
And like what d- what, what
combination of those rows
00:27:58.320 --> 00:28:01.070
does give zero.
00:28:01.070 --> 00:28:05.980
Suppose I add rows
one and rows three.
00:28:05.980 --> 00:28:08.660
If I add rows one and
rows three, what do I get?
00:28:08.660 --> 00:28:10.990
I get a row of all ones.
00:28:10.990 --> 00:28:15.340
Then if I add rows two and rows
four I get a row of all ones.
00:28:15.340 --> 00:28:19.370
So row one minus row two
plus row three minus row four
00:28:19.370 --> 00:28:21.280
is probably the zero row.
00:28:21.280 --> 00:28:24.490
It's a singular matrix.
00:28:24.490 --> 00:28:27.330
And I could find something in
its null space the same way.
00:28:27.330 --> 00:28:29.720
That would be a combination
of columns that gives zero.
00:28:29.720 --> 00:28:32.040
OK, there's an example.
00:28:32.040 --> 00:28:32.950
All right.
00:28:32.950 --> 00:28:36.840
So that's, well, that
shows two things.
00:28:36.840 --> 00:28:39.280
That shows how we get
the twenty-four terms
00:28:39.280 --> 00:28:41.700
and it shows the great
advantage of having
00:28:41.700 --> 00:28:43.810
a lot of zeros in there.
00:28:43.810 --> 00:28:45.830
OK.
00:28:45.830 --> 00:28:48.970
So we'll use this big
formula, but I want to pick --
00:28:48.970 --> 00:28:53.710
I want to go onward
now to cofactors.
00:28:53.710 --> 00:28:56.030
Onward to cofactors.
00:28:56.030 --> 00:29:03.330
Cofactors is a way of breaking
up this big formula that
00:29:03.330 --> 00:29:08.910
connects this n by n -- this
is an n by n determinant that
00:29:08.910 --> 00:29:13.880
we've just have a formula
for, the big formula.
00:29:13.880 --> 00:29:18.450
So cofactors is a way to connect
this n by n determinant to,
00:29:18.450 --> 00:29:22.270
determinants one smaller.
00:29:22.270 --> 00:29:24.170
One smaller.
00:29:24.170 --> 00:29:29.430
And the way we want to do it
is actually going to show up in
00:29:29.430 --> 00:29:30.260
this.
00:29:30.260 --> 00:29:34.470
Since the three by three is the
one that we wrote out in full,
00:29:34.470 --> 00:29:38.060
let's, let me do
this three by --
00:29:38.060 --> 00:29:43.350
so I'm talking about cofactors,
and I'm going to start again
00:29:43.350 --> 00:29:44.420
with three by three.
00:29:48.180 --> 00:29:51.220
And I'm going to take
the, the exact formula,
00:29:51.220 --> 00:29:55.690
and I'm just going to
write it as a one one --
00:29:55.690 --> 00:29:59.540
this is the determinant
I'm writing.
00:29:59.540 --> 00:30:03.360
I'm just going to say
a one one times what?
00:30:03.360 --> 00:30:04.760
A one one times what?
00:30:04.760 --> 00:30:08.830
And it's a one one times
a two two a three three
00:30:08.830 --> 00:30:12.230
minus a two three a three two.
00:30:15.410 --> 00:30:22.640
Then I've got the a one
two stuff times something.
00:30:22.640 --> 00:30:26.910
And I've got the a one
three stuff times something.
00:30:26.910 --> 00:30:29.500
Do you see what I'm doing?
00:30:29.500 --> 00:30:33.990
I'm taking our big formula
and I'm saying, OK,
00:30:33.990 --> 00:30:37.320
choose column --
00:30:37.320 --> 00:30:40.380
out of the first row,
choose column one.
00:30:40.380 --> 00:30:43.520
And take all the possibilities.
00:30:43.520 --> 00:30:46.240
And those extra
factors will be what
00:30:46.240 --> 00:30:51.800
we'll call the cofactor, co
meaning going with a one one.
00:30:51.800 --> 00:30:55.890
So this in parenthesis
are, these are in,
00:30:55.890 --> 00:30:57.930
the cofactors are in parens.
00:31:02.210 --> 00:31:05.660
A one one times something.
00:31:05.660 --> 00:31:09.820
And I figured out what that
something was by just looking
00:31:09.820 --> 00:31:14.290
back -- if I can walk back
here to the, to the a one one,
00:31:14.290 --> 00:31:17.440
the one that comes down the
diagonal minus the one that
00:31:17.440 --> 00:31:19.830
comes that way.
00:31:19.830 --> 00:31:24.570
That's, those are the two,
only two that used a one one.
00:31:24.570 --> 00:31:26.970
So there they are, one
with a plus and one with a
00:31:26.970 --> 00:31:28.240
minus.
00:31:28.240 --> 00:31:30.800
And now I can write in the --
00:31:30.800 --> 00:31:33.100
I could look back and
see what used a one two
00:31:33.100 --> 00:31:34.740
and I can see what
used a one three,
00:31:34.740 --> 00:31:36.770
and those will give
me the cofactors
00:31:36.770 --> 00:31:38.920
of a one two and a one
00:31:38.920 --> 00:31:41.650
three.
00:31:41.650 --> 00:31:45.770
Before I do that, what's this
number, what is this cofactor?
00:31:48.310 --> 00:31:52.080
What is it there that's
multiplying a one one?
00:31:52.080 --> 00:31:55.360
Tell me what a two two a three
three minus a two three a three
00:31:55.360 --> 00:31:59.310
two is, for this --
00:31:59.310 --> 00:32:00.340
do you recognize that?
00:32:03.600 --> 00:32:06.500
Do you recognize --
00:32:06.500 --> 00:32:08.780
let's see, I can --
and I'll put it here.
00:32:08.780 --> 00:32:11.520
There's the a one one.
00:32:11.520 --> 00:32:14.160
That's used column one.
00:32:14.160 --> 00:32:17.645
Then there's -- the other
factors involved these other
00:32:17.645 --> 00:32:18.145
columns.
00:32:25.070 --> 00:32:27.360
This row is used.
00:32:27.360 --> 00:32:29.140
This column is used.
00:32:29.140 --> 00:32:32.870
So this the only things
left to use are these.
00:32:32.870 --> 00:32:36.250
And this formula
uses them, and what's
00:32:36.250 --> 00:32:39.860
the, what's the cofactor?
00:32:39.860 --> 00:32:42.420
Tell me what it is because
you see it, and then --
00:32:42.420 --> 00:32:47.150
I'll be happy you see what
the idea of cofactors.
00:32:47.150 --> 00:32:50.010
It's the determinant
of this smaller guy.
00:32:52.920 --> 00:32:55.400
A one one multiplies
the determinant
00:32:55.400 --> 00:32:56.590
of this smaller guy.
00:32:56.590 --> 00:33:02.840
That gives me all the a one
one part of the big formula.
00:33:02.840 --> 00:33:03.540
You see that?
00:33:03.540 --> 00:33:05.500
This, the determinant
of this smaller guy
00:33:05.500 --> 00:33:10.540
is a two two a three three
minus a two three a three two.
00:33:10.540 --> 00:33:14.050
In other words, once I've
used column one and row
00:33:14.050 --> 00:33:20.850
one, what's left is all the
ways to use the other n-1
00:33:20.850 --> 00:33:25.130
columns and n-1
rows, one of each.
00:33:25.130 --> 00:33:28.970
All the other -- and that's the
determinant of the smaller guy
00:33:28.970 --> 00:33:30.880
of size n-1.
00:33:30.880 --> 00:33:33.710
So that's the whole
idea of cofactors.
00:33:33.710 --> 00:33:37.150
And we just have to remember
that with determinants we've
00:33:37.150 --> 00:33:39.900
got pluses and minus
signs to keep straight.
00:33:39.900 --> 00:33:43.260
Can we keep this
next one straight?
00:33:43.260 --> 00:33:45.090
Let's do the next one.
00:33:45.090 --> 00:33:51.640
OK, the next one will
be when I use a one two.
00:33:51.640 --> 00:33:55.330
I'll have left -- so I can't
use that column any more,
00:33:55.330 --> 00:34:02.710
but I can use a two one and a
two three and I can use a three
00:34:02.710 --> 00:34:06.330
one and a three three.
00:34:06.330 --> 00:34:08.850
So this one gave me a one
times that determinant.
00:34:08.850 --> 00:34:13.760
This will give me a one two
times this determinant, a two
00:34:13.760 --> 00:34:24.690
one a three three minus
a two three a three one.
00:34:24.690 --> 00:34:27.969
So that's all the stuff
involving a one two.
00:34:27.969 --> 00:34:32.360
But have I got the sign right?
00:34:32.360 --> 00:34:35.280
Is the determinant of that
correctly given by that
00:34:35.280 --> 00:34:37.510
or is there a minus sign?
00:34:37.510 --> 00:34:39.520
There is a minus sign.
00:34:39.520 --> 00:34:40.929
I can follow one of these.
00:34:40.929 --> 00:34:43.080
If I do that times
that times that,
00:34:43.080 --> 00:34:45.046
that was one that's
showing up here,
00:34:45.046 --> 00:34:47.420
but it should have showed --
it should have been a minus.
00:34:52.969 --> 00:34:55.675
So I'm going to build that
minus sign into the cofactor.
00:34:58.580 --> 00:35:01.390
So, so the cofactor
-- so I'll put,
00:35:01.390 --> 00:35:04.110
put that minus sign in here.
00:35:04.110 --> 00:35:07.180
So because the cofactor
is going to be strictly
00:35:07.180 --> 00:35:09.900
the thing that multiplies
the, the factor.
00:35:09.900 --> 00:35:12.570
The factor is a one two,
the cofactor is this,
00:35:12.570 --> 00:35:15.620
is the parens, the
stuff in parentheses.
00:35:15.620 --> 00:35:18.590
So it's got the
minus sign built in.
00:35:18.590 --> 00:35:23.960
And if I did -- if I went on
to the third guy, there w-
00:35:23.960 --> 00:35:26.920
there'll be this and
this, this and this.
00:35:26.920 --> 00:35:28.410
And it would take
its determinant.
00:35:28.410 --> 00:35:31.790
It would come out
plus the determinant.
00:35:31.790 --> 00:35:34.960
So now I'm ready to
say what cofactors are.
00:35:34.960 --> 00:35:40.620
So this would be a plus and a
one three times its cofactor.
00:35:40.620 --> 00:35:45.940
And over here we had plus a
one one times this determinant.
00:35:45.940 --> 00:35:49.400
But and there we had the a
one two times its cofactor,
00:35:49.400 --> 00:35:53.310
but the -- so the point is
the cofactor is either plus
00:35:53.310 --> 00:35:56.320
or minus the determinant.
00:35:56.320 --> 00:35:57.990
So let me write that
underneath them.
00:35:57.990 --> 00:36:00.750
What is the, what are cofactors?
00:36:00.750 --> 00:36:09.650
The cofactor if any
number aij, let's say.
00:36:13.500 --> 00:36:18.630
This is, this is all the terms
in the, in the big formula that
00:36:18.630 --> 00:36:20.260
involve aij.
00:36:20.260 --> 00:36:25.490
We're especially interested
in a1j, the first row, that's
00:36:25.490 --> 00:36:28.790
what I've been talking about,
but any row would be all right.
00:36:28.790 --> 00:36:30.650
All right, so --
00:36:30.650 --> 00:36:32.900
what terms involve aij?
00:36:32.900 --> 00:36:42.563
So -- it's the determinant
of the n minus one matrix --
00:36:46.430 --> 00:36:51.470
with row i, column j erased.
00:36:56.460 --> 00:37:00.900
So it's the, it's a
matrix of size n-1 with --
00:37:00.900 --> 00:37:05.710
of course, because I can't use
this row or this column again.
00:37:05.710 --> 00:37:08.120
So I have the matrix all there.
00:37:08.120 --> 00:37:11.400
But now it's multiplied
by a plus or a minus.
00:37:11.400 --> 00:37:14.165
This is the cofactor, and
I'm going to call that cij.
00:37:17.750 --> 00:37:19.990
Capital, I use
capital c just to,
00:37:19.990 --> 00:37:22.570
just to emphasize that
these are important
00:37:22.570 --> 00:37:25.850
and emphasize that
they're, they're, they're
00:37:25.850 --> 00:37:29.591
different from the (a)s.
00:37:29.591 --> 00:37:30.090
OK.
00:37:30.090 --> 00:37:34.340
So now is it a plus
or is it a minus?
00:37:34.340 --> 00:37:36.340
Because we see
that in this case,
00:37:36.340 --> 00:37:41.260
for a one one it was a plus,
for a one two I -- this is ij --
00:37:41.260 --> 00:37:43.360
it was a minus.
00:37:43.360 --> 00:37:46.530
For this ij it was a plus.
00:37:46.530 --> 00:37:50.550
So any any guess on the
rule for plus or minus
00:37:50.550 --> 00:37:55.650
when we see those examples,
ij equal one one or one three
00:37:55.650 --> 00:37:58.070
was a plus?
00:37:58.070 --> 00:38:04.400
It sounds very like
i+j odd or even.
00:38:04.400 --> 00:38:06.170
That, that's
doesn't surprise us,
00:38:06.170 --> 00:38:07.600
and that's the right answer.
00:38:07.600 --> 00:38:17.950
So it's a plus if i+j is even
and it's a minus if i+j is odd.
00:38:24.450 --> 00:38:28.170
So if I go along row one
and look at the cofactors,
00:38:28.170 --> 00:38:32.430
I just take those determinants,
those one smaller determinants,
00:38:32.430 --> 00:38:36.830
and they come in order plus
minus plus minus plus minus.
00:38:36.830 --> 00:38:42.120
But if I go along row two and,
and, and take the cofactors
00:38:42.120 --> 00:38:46.310
of sub-determinants, they
would start with a minus,
00:38:46.310 --> 00:38:52.520
because the two one entry,
two plus one is odd, so the --
00:38:52.520 --> 00:38:57.560
like there's a pattern plus
minus plus minus plus if it was
00:38:57.560 --> 00:39:01.180
five by five, but then if I was
doing a cofactor then this sign
00:39:01.180 --> 00:39:06.087
would be minus plus minus
plus minus, plus minus plus --
00:39:06.087 --> 00:39:07.170
it's sort of checkerboard.
00:39:12.540 --> 00:39:13.040
OK.
00:39:17.551 --> 00:39:18.050
OK.
00:39:18.050 --> 00:39:22.220
Those are the signs that,
that are given by this rule,
00:39:22.220 --> 00:39:24.540
i+j even or odd.
00:39:24.540 --> 00:39:27.760
And those are built
into the cofactors.
00:39:27.760 --> 00:39:31.030
The thing is called
a minor without th-
00:39:31.030 --> 00:39:34.400
before you've built in the sign,
but I don't care about those.
00:39:34.400 --> 00:39:39.700
Build in that sign and
call it a cofactor.
00:39:39.700 --> 00:39:41.950
So what's the cofactor formula?
00:39:41.950 --> 00:39:42.450
OK.
00:39:42.450 --> 00:39:44.240
What's the cofactor
formula then?
00:39:44.240 --> 00:39:49.110
Let me come back to
this board and say,
00:39:49.110 --> 00:39:50.560
what's the cofactor formula?
00:39:58.150 --> 00:40:02.840
Determinant of A is --
00:40:02.840 --> 00:40:04.670
let's go along the first row.
00:40:04.670 --> 00:40:11.480
It's a one one
times its cofactor,
00:40:11.480 --> 00:40:15.730
and then the second guy is a
one two times its cofactor,
00:40:15.730 --> 00:40:19.070
and you just keep going
to the end of the row,
00:40:19.070 --> 00:40:22.750
a1n times its cofactor.
00:40:22.750 --> 00:40:24.840
So that's cofactor for --
00:40:24.840 --> 00:40:30.780
along row one.
00:40:30.780 --> 00:40:39.030
And if I went along row I, I
would -- those ones would be
00:40:39.030 --> 00:40:39.530
Is.
00:40:39.530 --> 00:40:43.520
That's worth putting a box over.
00:40:43.520 --> 00:40:47.470
That's the cofactor formula.
00:40:47.470 --> 00:40:51.180
Do you see that --
00:40:51.180 --> 00:40:53.740
actually, this would
give me another way
00:40:53.740 --> 00:41:00.150
I could have started the
whole topic of determinants.
00:41:00.150 --> 00:41:02.000
And some, some people
might do it this --
00:41:02.000 --> 00:41:04.440
choose to do it this way.
00:41:04.440 --> 00:41:06.540
Because the cofactor
formula would
00:41:06.540 --> 00:41:09.740
allow me to build up an
n by n determinant out
00:41:09.740 --> 00:41:14.790
of n-1 sized determinants, build
those out of n-2, and so on.
00:41:14.790 --> 00:41:17.740
I could boil all the
way down to one by ones.
00:41:17.740 --> 00:41:21.470
So what's the cofactor formula
for two by two matrices?
00:41:21.470 --> 00:41:23.300
Yeah, tell me that.
00:41:23.300 --> 00:41:24.580
What's the cofactor for us?
00:41:24.580 --> 00:41:28.620
Here is the, here is the world's
smallest example, practically,
00:41:28.620 --> 00:41:32.580
of a cofactor formula.
00:41:32.580 --> 00:41:33.170
OK.
00:41:33.170 --> 00:41:35.570
Let's go along row one.
00:41:35.570 --> 00:41:39.500
I take this first guy
times its cofactor.
00:41:39.500 --> 00:41:44.350
What's the cofactor
of the one one entry?
00:41:44.350 --> 00:41:48.460
d, because you strike out
the one one row and column
00:41:48.460 --> 00:41:50.460
and you're left with d.
00:41:50.460 --> 00:41:54.330
Then I take this guy,
b, times its cofactor.
00:41:54.330 --> 00:41:57.740
What's the cofactor of b?
00:41:57.740 --> 00:42:00.070
Is it c or it's --
00:42:00.070 --> 00:42:03.330
minus c, because I
strike out this guy,
00:42:03.330 --> 00:42:08.360
I take that determinant, and
then I follow the i+j rule
00:42:08.360 --> 00:42:11.730
and I get a minus, I get an odd.
00:42:11.730 --> 00:42:13.450
So it's b times minus c.
00:42:17.230 --> 00:42:18.120
OK, it worked.
00:42:18.120 --> 00:42:20.430
Of course it, it worked.
00:42:20.430 --> 00:42:23.100
And the three by three works.
00:42:23.100 --> 00:42:28.200
So that's the cofactor formula,
and that is, that's an --
00:42:28.200 --> 00:42:33.230
that's a good formula to know,
and now I'm feeling like, wow,
00:42:33.230 --> 00:42:38.560
I'm giving you a lot of
algebra to swallow here.
00:42:38.560 --> 00:42:41.955
Last lecture gave
you ten properties.
00:42:44.900 --> 00:42:46.750
Now I'm giving you --
00:42:46.750 --> 00:42:50.390
and by the way, those ten
properties led us to a formula
00:42:50.390 --> 00:42:52.480
for the determinant
which was very important,
00:42:52.480 --> 00:42:55.850
and I haven't
repeated it till now.
00:42:55.850 --> 00:42:56.820
What was that?
00:42:56.820 --> 00:43:01.140
The, the determinant is
the product of the pivots.
00:43:01.140 --> 00:43:03.980
So the pivot formula
is, is very important.
00:43:03.980 --> 00:43:07.440
The pivots have all this
complicated mess already
00:43:07.440 --> 00:43:08.810
built in.
00:43:08.810 --> 00:43:11.460
As you did elimination
to get the pivots,
00:43:11.460 --> 00:43:17.670
you built in all this horrible
stuff, quite efficiently.
00:43:17.670 --> 00:43:20.680
Then the big formula with
the n factorial terms,
00:43:20.680 --> 00:43:24.180
that's got all the
horrible stuff spread out.
00:43:24.180 --> 00:43:28.870
And the cofactor formula
is like in between.
00:43:28.870 --> 00:43:35.460
It's got easy stuff times
horrible stuff, basically.
00:43:35.460 --> 00:43:39.840
But it's, it shows you,
how to get determinants
00:43:39.840 --> 00:43:42.780
from smaller determinants, and
that's the application that I
00:43:42.780 --> 00:43:45.290
now want to make.
00:43:45.290 --> 00:43:51.030
So may I do one more example?
00:43:51.030 --> 00:43:54.670
So I remember the general idea.
00:43:54.670 --> 00:43:59.360
But I'm going to use this
cofactor formula for a matrix
00:43:59.360 --> 00:44:01.230
--
00:44:01.230 --> 00:44:03.640
so here is going
to be my example.
00:44:03.640 --> 00:44:07.890
It's -- I promised in
the, in the lecture,
00:44:07.890 --> 00:44:11.090
outline at the very
beginning to do an example.
00:44:11.090 --> 00:44:12.650
And let me do --
00:44:12.650 --> 00:44:18.280
I'm going to pick
tri-diagonal matrix of ones.
00:44:21.700 --> 00:44:26.340
I could, I'm drawing
here the four by four.
00:44:26.340 --> 00:44:27.880
So this will be the matrix.
00:44:27.880 --> 00:44:29.260
I could call that A4.
00:44:34.830 --> 00:44:40.590
But my real idea
is to do n by n.
00:44:40.590 --> 00:44:43.100
To do them all.
00:44:43.100 --> 00:44:44.930
So A -- I could --
00:44:44.930 --> 00:44:49.010
everybody understands
what A1 and A2 are.
00:44:49.010 --> 00:44:49.510
Yeah.
00:44:49.510 --> 00:44:54.900
Maybe we should just do A1
and A2 and A3 just for --
00:44:54.900 --> 00:44:55.400
so this is
00:44:55.400 --> 00:44:57.591
What's the determinant of A1?
00:44:57.591 --> 00:44:58.090
A4.
00:44:58.090 --> 00:45:01.120
What's the determinant of A1?
00:45:01.120 --> 00:45:04.470
So, so what's the matrix
A1 in this formula?
00:45:04.470 --> 00:45:06.310
It's just got that.
00:45:06.310 --> 00:45:08.850
So the determinant is one.
00:45:08.850 --> 00:45:10.950
What's the determinant of A2?
00:45:10.950 --> 00:45:14.831
So it's just got this two by
two, and its determinant is --
00:45:17.780 --> 00:45:19.690
zero.
00:45:19.690 --> 00:45:22.350
And then the three by three.
00:45:22.350 --> 00:45:23.800
Can we see its determinant?
00:45:23.800 --> 00:45:27.740
Can you take the determinant
of that three by three?
00:45:27.740 --> 00:45:32.910
Well, that's not quite so
obvious, at least not to me.
00:45:32.910 --> 00:45:35.060
Being three by three,
I don't know --
00:45:35.060 --> 00:45:36.710
so here's a, here's
a good example.
00:45:36.710 --> 00:45:39.590
How would you do that
three by three determinant?
00:45:39.590 --> 00:45:43.160
We've got, like, n
factorial different ways.
00:45:43.160 --> 00:45:44.080
Well, three factorial.
00:45:44.080 --> 00:45:45.260
So we've got six ways.
00:45:45.260 --> 00:45:46.220
OK.
00:45:46.220 --> 00:45:49.440
I mean, one way to do it --
00:45:49.440 --> 00:45:51.040
actually the way
I would probably
00:45:51.040 --> 00:45:52.990
do it, being three
by three, I would use
00:45:52.990 --> 00:45:55.250
the complete the big formula.
00:45:55.250 --> 00:45:57.120
I would say, I've
got a one from that,
00:45:57.120 --> 00:46:00.710
I've got a zero from that, I've
got a zero from that, a zero
00:46:00.710 --> 00:46:03.140
from that, and this
direction is a minus one,
00:46:03.140 --> 00:46:04.410
that direction's a minus one.
00:46:04.410 --> 00:46:06.200
I believe the
answer is minus one.
00:46:06.200 --> 00:46:14.220
Would you do it another way?
00:46:14.220 --> 00:46:16.400
Here's another way
to do it, look.
00:46:16.400 --> 00:46:18.065
Subtract row three from --
00:46:18.065 --> 00:46:19.440
I'm just looking
at this three by
00:46:19.440 --> 00:46:19.939
three.
00:46:19.939 --> 00:46:22.400
Everybody's looking
at the three by three.
00:46:22.400 --> 00:46:25.630
Subtract row three from row two.
00:46:25.630 --> 00:46:26.930
Determinant doesn't change.
00:46:26.930 --> 00:46:29.710
So those become zeros.
00:46:29.710 --> 00:46:31.710
OK, now use the
cofactor formula.
00:46:31.710 --> 00:46:33.160
How's that?
00:46:33.160 --> 00:46:36.310
How can, how -- if this was now
zeros and I'm looking at this
00:46:36.310 --> 00:46:39.530
three by three, use
the cofactor formula.
00:46:39.530 --> 00:46:42.875
Why not use the cofactor
formula along that row?
00:46:45.890 --> 00:46:49.360
Because then I take that
number times its cofactor,
00:46:49.360 --> 00:46:52.800
so I take this number -- let
me put a box around it --
00:46:52.800 --> 00:46:56.030
times its cofactor, which is the
determinant of that and that,
00:46:56.030 --> 00:46:56.830
which is what?
00:47:02.310 --> 00:47:07.230
That two by two matrix
has determinant one.
00:47:07.230 --> 00:47:08.625
So what's the cofactor?
00:47:11.410 --> 00:47:15.150
What's the cofactor
of this guy here?
00:47:15.150 --> 00:47:17.220
Looking just at
this three by three.
00:47:17.220 --> 00:47:21.470
The cofactor of that
one is this determinant,
00:47:21.470 --> 00:47:26.490
which is one times negative.
00:47:26.490 --> 00:47:30.370
So that's why the answer
came out minus one.
00:47:30.370 --> 00:47:31.280
OK.
00:47:31.280 --> 00:47:32.910
So I did the three by three.
00:47:32.910 --> 00:47:35.310
I don't know if we want
to try the four by four.
00:47:35.310 --> 00:47:38.090
Yeah, let's -- I guess that
was the point of my example,
00:47:38.090 --> 00:47:41.250
of course, so I have to try it.
00:47:41.250 --> 00:47:44.120
Sorry, I'm in a good
mood today, so you have
00:47:44.120 --> 00:47:45.840
to stand for all the bad jokes.
00:47:45.840 --> 00:47:46.340
OK.
00:47:46.340 --> 00:47:47.200
OK.
00:47:47.200 --> 00:47:50.830
So what was the matrix?
00:47:50.830 --> 00:47:51.330
Ah.
00:47:55.580 --> 00:47:57.660
OK, now I'm ready
for four by four.
00:47:57.660 --> 00:48:00.720
Who wants to -- who wants
to guess the, the --
00:48:00.720 --> 00:48:04.270
I don't know, frankly,
this four by four,
00:48:04.270 --> 00:48:07.280
what's, what's the determinant.
00:48:07.280 --> 00:48:08.575
I plan to use cofactors.
00:48:12.640 --> 00:48:14.310
OK, let's use cofactors.
00:48:14.310 --> 00:48:17.960
The determinant of A4 is --
00:48:17.960 --> 00:48:20.420
OK, let's use cofactors
on the first row.
00:48:20.420 --> 00:48:21.630
Those are easy.
00:48:21.630 --> 00:48:25.720
So I multiply this number,
which is a convenient one,
00:48:25.720 --> 00:48:28.240
times this determinant.
00:48:28.240 --> 00:48:31.790
So it's, it's one times
the, this three by three
00:48:31.790 --> 00:48:32.380
determinant.
00:48:32.380 --> 00:48:36.600
Now what is -- do you
recognize that matrix?
00:48:36.600 --> 00:48:37.900
It's A3.
00:48:37.900 --> 00:48:41.940
So it's one times the
determinant of A3.
00:48:41.940 --> 00:48:46.830
Coming along this row is a
one times this determinant,
00:48:46.830 --> 00:48:49.430
and it goes with a plus, right?
00:48:49.430 --> 00:48:50.665
And then we have this one.
00:48:53.170 --> 00:48:55.290
And what is its cofactor?
00:48:55.290 --> 00:48:58.780
Now I'm looking at, now
I'm looking at this three
00:48:58.780 --> 00:49:00.990
by three, this three
by three, so I'm
00:49:00.990 --> 00:49:04.270
looking at the three by three
that I haven't X-ed out.
00:49:04.270 --> 00:49:08.630
What is that -- oh, now
it, we did a plus or a --
00:49:08.630 --> 00:49:10.880
is it plus this determinant,
this three by three
00:49:10.880 --> 00:49:13.730
determinant, or minus it?
00:49:13.730 --> 00:49:16.150
It's minus it, right,
because this is --
00:49:16.150 --> 00:49:20.180
I'm starting in a one two
position, and that's a minus.
00:49:20.180 --> 00:49:22.390
So I want minus
this determinant.
00:49:22.390 --> 00:49:25.180
But these guys are X-ed out.
00:49:25.180 --> 00:49:25.680
OK.
00:49:25.680 --> 00:49:27.260
So I've got a three by three.
00:49:27.260 --> 00:49:31.170
Well, let's use cofactors again.
00:49:31.170 --> 00:49:33.740
Use cofactors of the
column, because actually we
00:49:33.740 --> 00:49:35.620
could use cofactors
of columns just
00:49:35.620 --> 00:49:39.930
as well as rows, because,
because the transpose rule.
00:49:39.930 --> 00:49:43.750
So I'll take this one, which
is now sitting in the plus
00:49:43.750 --> 00:49:46.440
position, times its
determinant -- oh!
00:49:46.440 --> 00:49:47.773
Oh, hell.
00:49:50.870 --> 00:49:53.990
What -- oh yeah, I
shouldn't have said hell,
00:49:53.990 --> 00:49:55.260
because it's all right.
00:49:55.260 --> 00:49:55.760
OK.
00:49:55.760 --> 00:49:58.230
One times the determinant.
00:49:58.230 --> 00:50:00.410
What is that matrix
now that I'm taking
00:50:00.410 --> 00:50:01.920
the, this smaller one of?
00:50:01.920 --> 00:50:03.420
Oh, but there's a minus, right?
00:50:03.420 --> 00:50:05.630
The minus came
from, from the fact
00:50:05.630 --> 00:50:10.570
that this was in the one
two position and that's odd.
00:50:10.570 --> 00:50:14.750
So this is a minus one
times -- and what's --
00:50:14.750 --> 00:50:17.130
and then this one
is the upper left,
00:50:17.130 --> 00:50:21.880
that's the one one position
in its matrix, so plus.
00:50:21.880 --> 00:50:23.910
And what's this matrix here?
00:50:23.910 --> 00:50:26.460
Do you recognize that?
00:50:26.460 --> 00:50:28.840
That matrix is --
00:50:28.840 --> 00:50:30.275
yes, please say it --
00:50:30.275 --> 00:50:30.775
A2.
00:50:36.320 --> 00:50:39.110
And we -- that's our
formula for any case.
00:50:39.110 --> 00:50:46.790
A of any size n is equal to the
determinant of A n minus one,
00:50:46.790 --> 00:50:49.310
that's what came from taking
the one in the upper corner,
00:50:49.310 --> 00:50:55.520
the first cofactor, minus the
determinant of A n minus two.
00:50:55.520 --> 00:51:02.020
What we discovered
there is true for all n.
00:51:02.020 --> 00:51:04.680
I didn't even mention
it, but I stopped taking
00:51:04.680 --> 00:51:06.610
cofactors when I got this one.
00:51:06.610 --> 00:51:08.920
Why did I stop?
00:51:08.920 --> 00:51:12.660
Why didn't I take the
cofactor of this guy?
00:51:12.660 --> 00:51:16.490
Because he's going to get
multiplied by zero, and no,
00:51:16.490 --> 00:51:18.130
no use wasting time.
00:51:18.130 --> 00:51:20.260
Or this one too.
00:51:20.260 --> 00:51:22.570
The cofactor, her
cofactor will be
00:51:22.570 --> 00:51:24.400
whatever that
determinant is, but it'll
00:51:24.400 --> 00:51:26.980
be multiplied by zero,
so I won't bother.
00:51:26.980 --> 00:51:30.010
OK, there is the formula.
00:51:30.010 --> 00:51:32.160
And that now tells us --
00:51:32.160 --> 00:51:34.620
I could figure out
what A4 is now.
00:51:34.620 --> 00:51:37.570
Oh yeah, finally I can get A4.
00:51:37.570 --> 00:51:43.770
Because it's A3, which is minus
one, minus A2, which is zero,
00:51:43.770 --> 00:51:44.880
so it's minus one.
00:51:47.750 --> 00:51:50.090
You see how we're
getting kind of numbers
00:51:50.090 --> 00:51:51.710
that you might not have guessed.
00:51:51.710 --> 00:51:55.800
So our sequence right now is
one zero minus one minus one.
00:51:55.800 --> 00:52:00.760
What's the next one
in the sequence, A5?
00:52:00.760 --> 00:52:06.300
A5 is this minus
this, so it is zero.
00:52:06.300 --> 00:52:09.260
What's A6?
00:52:09.260 --> 00:52:15.140
A6 is this minus
this, which is one.
00:52:15.140 --> 00:52:18.150
What's A7?
00:52:18.150 --> 00:52:21.640
I'm, I'm going to be stopped
by either the time runs out
00:52:21.640 --> 00:52:23.230
or the board runs out.
00:52:23.230 --> 00:52:27.740
OK, A7 is this minus
this, which is one.
00:52:27.740 --> 00:52:30.790
I'll stop here, because time
is out, but let me tell you
00:52:30.790 --> 00:52:32.180
what we've got.
00:52:32.180 --> 00:52:35.850
What -- these determinants
have this series,
00:52:35.850 --> 00:52:39.730
one zero minus one
minus one zero one,
00:52:39.730 --> 00:52:42.760
and then it starts repeating.
00:52:42.760 --> 00:52:44.600
It's pretty fantastic.
00:52:44.600 --> 00:52:48.480
These determinants
have period six.
00:52:48.480 --> 00:52:51.880
So the determinant
of A sixty-one
00:52:51.880 --> 00:52:55.771
would be the determinant
of A1, which would be one.
00:52:55.771 --> 00:52:56.270
OK.
00:52:56.270 --> 00:52:58.190
I hope you liked that example.
00:52:58.190 --> 00:53:04.640
A non-trivial example of a
tri-diagonal determinant.
00:53:04.640 --> 00:53:05.520
Thanks.
00:53:05.520 --> 00:53:08.260
See you on Wednesday.