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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 inside
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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,
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but they --
those changed,
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that situation changed.
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Now determinants is one
specific part,
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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 this number,
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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 for some,
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some matrices that have a
lot of zeros away from the
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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,
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three simple properties,
but out of that we got all
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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 -- oh,
how about I just take two by
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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
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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
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identity.
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Right?
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Property two allows me to
exchange
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rows, 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 that this
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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,
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linearity 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 --
if I can make the matrices
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diagonal, 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 into a c 0 and,
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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
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more 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 -- well,
all four are extremely easy.
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Two of them are so easy as to
turn into zero,
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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,
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well, 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,
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then -- so I,
I'm finally getting to that
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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 to
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make it c b, then property three
to factor out the b,
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property c to factor out the c
-- the property again to factor
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out the c, and that minus,
and of course finally I got the
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answer that we knew we would
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
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for here, 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 -- so
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let's -- what would happen with
three by threes?
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Can we mentally do it rather
than I write everything on the
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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 and I would split the
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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
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zero 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
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four pieces, I would go from one
piece to three pieces to -- what
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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
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out, 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 -- separating out
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one row into three pieces would
give me three,
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separating out the second row
into three pieces,
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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,
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three cubed,
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 using these double
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symbols here because otherwise I
could never 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 is going to be what are
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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 --
for example, one survivor will
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be 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.
a one one zero zero zero a two
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two zero zero 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 -- the
survivors have one entry from
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each row and each column.
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One entry from each row and
column.
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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
-- 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
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just put -- 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
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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?
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Use column three,
because here if I use column --
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here I used column one and row
one.
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This was like the column --
numbers were one two three,
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right in order.
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Now the column numbers are
going to
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be one three,
column three,
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and column two.
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So the row numbers are one two
three, of course.
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The column numbers are some --
OK, some permutation of one two
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three, and here they come in the
order one three two.
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It's just like having a
permutation matrix with,
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instead of the ones,
with numbers.
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And actually,
it's very close to having a
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permutation matrix,
because I, what I do eventually
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is I factor out these numbers
and then I have got.
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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.
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And what does this determinant
equal?
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Yeah, now tell me the,
this -- I mean,
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we're really getting to the
heart of these formulas now.
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What is that determinant?
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By the laws of -- by,
by our three properties,
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I can factor these out,
I can factor out the a one one,
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the a two three,
and the a three two.
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They're in separate rows.
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I can do each row separately.
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And then I just have to decide
is that plus sign or is that a
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minus sign?
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And the answer is it's a minus.
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Why minus?
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Because these is one row
exchange to get it back to the
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identity.
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So that's a minus.
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Now I through?
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No, because there are other
ways.
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What I'm really through with,
what I've done,
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what I've, what I've completed
is only the part where the a one
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one is there.
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But now I've got parts
where it's a one two.
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And now if it's a one two that
row is used, that column is
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used.
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You see that idea?
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I could use this row and
column.
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Now that column is used,
that column is used,
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and this guy has to be here,
a three three.
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And what's that determinant?
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That's an a one two times an a
two one times an a three three,
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and does it have a plus or a
minus?
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A minus is right.
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It has a minus.
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Because it's one flip away from
an id- from the,
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regular, the right order,
the diagonal order.
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And now what's the other guy
with a --
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with, a one two up there?
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I could have used this row.
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I could have put this guy here
and this guy here.
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Right?
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You see the whole deal?
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Now that's an a one two,
a two three,
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a three one,
and does that go with a plus or
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a minus?
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Yeah, now that takes a minute
of thinking, doesn't it,
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because one row exchange
doesn't get it in line.
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So what is the answer for this?
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Plus or minus?
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Plus, because it takes two
exchanges.
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I could exchange rows one and
three and then two and three.
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Two exchanges makes this thing
a plus.
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OK.
251
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And then finally we have --
we're going to have two more.
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Zero zero a one three,
a two one zero zero,
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zero a three two zero.
254
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And one more guy.
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Zero zero a one three,
zero a two two zero,
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A three one zero zero.
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And let's put down what we get
from those.
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An a one three,
an a two one,
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and an a three two,
and I think that one is a plus.
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And this guys is a minus
because one exchange would put
261
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it -- would order it.
262
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And that's a minus.
263
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All right, that has taken one
whole board just to do the three
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by three.
265
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But do you agree that we now
have a formula for the
266
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determinant which came from the
three properties?
267
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And it must be it.
268
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And I'm going to keep that
formula.
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That's a famous --
that three by three formula is
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one that if, if the cameras will
follow me back to the beginning
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here, I, I get the ones with the
plus sign are the ones that go
272
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down like down this way.
273
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And the ones with the minus
signs are sort of the ones that
274
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go this way.
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I won't make that precise.
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For two reasons,
one, it would clutter up the
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board, and second reason,
it wouldn't be right for four
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00:15:28 --> 00:15:28
by fours.
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For four by four,
let me just say right away,
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four by four matrix -- the,
the cross diagonal,
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the wrong diagonal happens to
come out with a plus sign.
282
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Why is that?
283
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If I have a four by four matrix
with ones coming on the counter
284
00:15:48 --> 00:15:52
diagonal, that determinant is
plus.
285
00:15:52 --> 00:15:52
Why?
286
00:15:52 --> 00:15:55
Why plus for that guy?
287
00:15:55 --> 00:16:01.08
Because if I exchange rows one
and four and then I exchange
288
00:16:01.08 --> 00:16:06
rows two and three,
I've got the identity,
289
00:16:06 --> 00:16:09
and I did two exchanges.
290
00:16:09 --> 00:16:13
So this down to this,
like, you know,
291
00:16:13 --> 00:16:18
down toward Miami and down
toward LA stuff is,
292
00:16:18 --> 00:16:22
like, three by three only.
293
00:16:22 --> 00:16:22
OK.
294
00:16:22 --> 00:16:29
But I do want to get now -- I
don't want to go through this
295
00:16:29 --> 00:16:31
for a four by four.
296
00:16:31 --> 00:16:37
I do want to get now
the general formula.
297
00:16:37 --> 00:16:43.79
So this is what I refer to in
the book as the big formula.
298
00:16:43.79 --> 00:16:49
So now this is the big formula
for the determinant.
299
00:16:49 --> 00:16:55.74
I'm asking you to make a jump
from two by two and three by
300
00:16:55.74 --> 00:16:57
three to n by n.
301
00:16:57 --> 00:17:01
OK, so this will be the big
formula.
302
00:17:01 --> 00:17:06
That the determinant of A is
the sum
303
00:17:06 --> 00:17:08.75
of a whole lot of terms.
304
00:17:08.75 --> 00:17:10.9
And what are those terms?
305
00:17:10.9 --> 00:17:15
And, and is it a plus or a
minus sign, and I have to tell
306
00:17:15 --> 00:17:20
you which, which it is,
because this came in -- in the
307
00:17:20 --> 00:17:23
three by three case,
I had how many terms?
308
00:17:23 --> 00:17:24
Six.
309
00:17:24 --> 00:17:28
And half were plus and half
were minus.
310
00:17:28 --> 00:17:33
How many terms are you figuring
for four by four?
311
00:17:33 --> 00:17:40
If I get two terms in the two
by two case, three -- six terms
312
00:17:40 --> 00:17:46
in the three by three case,
what's that pattern?
313
00:17:46 --> 00:17:50
How many terms in the four by
four case?
314
00:17:50 --> 00:17:51
Twenty-four.
315
00:17:51 --> 00:17:54
Four factorial.
316
00:17:54 --> 00:17:56
Why four factorial?
317
00:17:56 --> 00:18:00.79
This will be a sum of n
factorial terms.
318
00:18:00.79 --> 00:18:04
Twenty-four,
a hundred and twenty,
319
00:18:04 --> 00:18:09
seven hundred and twenty,
whatever's after that.
320
00:18:09 --> 00:18:09
OK.
321
00:18:09 --> 00:18:12
Half plus and half minus.
322
00:18:12 --> 00:18:16
And where do those n factorial
--
323
00:18:16 --> 00:18:17
terms come from?
324
00:18:17 --> 00:18:21
This is the moment to listen to
this lecture.
325
00:18:21 --> 00:18:24
Where do those n factorial
terms come from?
326
00:18:24 --> 00:18:29
They come because the first,
the guy in the first row can be
327
00:18:29 --> 00:18:31
chosen n ways.
328
00:18:31 --> 00:18:34
And after he's chosen,
that's used up that,
329
00:18:34 --> 00:18:36
that column.
330
00:18:36 --> 00:18:40
So the one in the second row
can be chosen n minus one ways.
331
00:18:40 --> 00:18:44
And after she's chosen,
that second column has been
332
00:18:44 --> 00:18:44
used.
333
00:18:44 --> 00:18:49.35
And then the one in the third
row can be chosen n minus two
334
00:18:49.35 --> 00:18:53
ways, and after it's chosen --
notice how I'm getting these
335
00:18:53 --> 00:18:55
personal pronouns.
336
00:18:55 --> 00:18:57
But I've run out.
337
00:18:57 --> 00:19:00
And I'm not willing to stop
with three by three,
338
00:19:00 --> 00:19:04
so I'm just going to write the
formula down.
339
00:19:04 --> 00:19:08.57
So the one in the first row
comes from some column alpha.
340
00:19:08.57 --> 00:19:10
I don't know what alpha is.
341
00:19:10 --> 00:19:14
And the one in the -- I
multiply that by somebody in the
342
00:19:14 --> 00:19:19.5
second row that comes from some
different column.
343
00:19:19.5 --> 00:19:24
And I multiply that by somebody
in the third row who comes from
344
00:19:24 --> 00:19:26
some yet different column.
345
00:19:26 --> 00:19:30
And then in the n-th row,
I don't know what -- I don't
346
00:19:30 --> 00:19:31
know how to draw.
347
00:19:31 --> 00:19:33
Maybe omega,
for last.
348
00:19:33 --> 00:19:38
And the whole point is then
that -- that those column
349
00:19:38 --> 00:19:42.14
numbers are different,
that alpha, beta,
350
00:19:42.14 --> 00:19:46
gamma, omega,
that set of column numbers is
351
00:19:46 --> 00:19:50
some permutation,
permutation of one to n.
352
00:19:50 --> 00:19:55
It, it, the n column numbers
are each used once.
353
00:19:55 --> 00:19:58
And that gives us n factorial
terms.
354
00:19:58 --> 00:20:03
And when I choose a term,
that means I'm choosing
355
00:20:03 --> 00:20:07
somebody
from every row and column.
356
00:20:07 --> 00:20:12
And then I just -- like the way
I had this from row and column
357
00:20:12 --> 00:20:16
one, row and column two,
row and column three,
358
00:20:16 --> 00:20:20
so that -- what was the alpha
beta stuff in that,
359
00:20:20 --> 00:20:21
for that term here?
360
00:20:21 --> 00:20:23
Alpha was one,
beta was two,
361
00:20:23 --> 00:20:25
gamma was three.
362
00:20:25 --> 00:20:30
The permutation was,
was the trivial permutation,
363
00:20:30 --> 00:20:33
one two three,
everybody in the right order.
364
00:20:33 --> 00:20:35.26
You see that formula?
365
00:20:35.26 --> 00:20:39
It's -- do you see why I didn't
want to start with that the
366
00:20:39 --> 00:20:41
first day, Friday?
367
00:20:41 --> 00:20:44
I'd rather we understood the
properties.
368
00:20:44 --> 00:20:50
Because out of this formula,
presumably I could figure out
369
00:20:50 --> 00:20:51.76
all these properties.
370
00:20:51.76 --> 00:20:56.03
How would I know that the
determinant of the identity
371
00:20:56.03 --> 00:20:58
matrix was one,
for example,
372
00:20:58 --> 00:20:59
out of this formula?
373
00:20:59 --> 00:21:04
Why is -- if A is the identity
matrix, how does this formula
374
00:21:04 --> 00:21:06
give me a plus one?
375
00:21:06 --> 00:21:08
You see it, right?
376
00:21:08 --> 00:21:12
Because, because almost all the
terms are zeros.
377
00:21:12 --> 00:21:15
Which term isn't zero,
if, if A is the identity
378
00:21:15 --> 00:21:16
matrix?
379
00:21:16 --> 00:21:20
Almost all the terms are zero
because almost all the As are
380
00:21:20 --> 00:21:20
zero.
381
00:21:20 --> 00:21:25
It's only, the only time I'll
get something is if it's a one
382
00:21:25 --> 00:21:28
one times a two two times a
three three.
383
00:21:28 --> 00:21:31
Only, only the,
only the permutation that's in
384
00:21:31 --> 00:21:34
the right order will,
will give me something.
385
00:21:34 --> 00:21:36
It'll come with a plus sign.
386
00:21:36 --> 00:21:39.7
And the determinant of the
identity is one.
387
00:21:39.7 --> 00:21:43
So, so we could go back from
this formula and prove
388
00:21:43 --> 00:21:43
everything.
389
00:21:43 --> 00:21:46
We could even try to prove that
the
390
00:21:46 --> 00:21:50
determinant of A B was the
determinant of A times the
391
00:21:50 --> 00:21:52
determinant of B.
392
00:21:52 --> 00:21:55
But like next week we would
still be working on it,
393
00:21:55 --> 00:21:59
because it's not -- clear from
-- if I took A B,
394
00:21:59 --> 00:21:59
my God.
395
00:21:59 --> 00:22:00
You know --.
396
00:22:00 --> 00:22:04
The entries of A B would
be all these pieces.
397
00:22:04 --> 00:22:08.25
Well, probably,
it's probably -- historically
398
00:22:08.25 --> 00:22:11.64
it's been done,
but it won't be repeated in
399
00:22:11.64 --> 00:22:12
eighteen oh six.
400
00:22:12 --> 00:22:13
OK.
401
00:22:13 --> 00:22:18
It would be possible probably
to see, why the determinant of A
402
00:22:18 --> 00:22:21
equals the
determinant of A transpose.
403
00:22:21 --> 00:22:24
That was another,
like, miracle property at the
404
00:22:24 --> 00:22:25
end.
405
00:22:25 --> 00:22:28
That would, that would,
that's an easier one,
406
00:22:28 --> 00:22:29
which we could find.
407
00:22:29 --> 00:22:29
OK.
408
00:22:29 --> 00:22:32
Is that all right for the big
formula?
409
00:22:32 --> 00:22:37.36
I could take you then a,
a typical -- let me do an
410
00:22:37.36 --> 00:22:38
example.
411
00:22:38 --> 00:22:40
Which I'll just create.
412
00:22:40 --> 00:22:43
I'll take a four by four
matrix.
413
00:22:43 --> 00:22:47
I'll put some,
I'll put some ones in and some
414
00:22:47 --> 00:22:48
zeros in.
415
00:22:48 --> 00:22:48
OK.
416
00:22:48 --> 00:22:53.13
Let me -- I don't know how many
to
417
00:22:53.13 --> 00:22:55
put in, to tell the truth.
418
00:22:55 --> 00:22:58
I've never done this before.
419
00:22:58 --> 00:23:02.93
I don't know the determinant of
that matrix.
420
00:23:02.93 --> 00:23:08
So like mathematics is being
done for the first time in,
421
00:23:08 --> 00:23:10
in front of your eyes.
422
00:23:10 --> 00:23:13
What's the determinant?
423
00:23:13 --> 00:23:18
Well, a lot of -- there are
twenty-four terms,
424
00:23:18 --> 00:23:20
because it's four by four.
425
00:23:20 --> 00:23:25
Many of them will be zero,
because I've got all those
426
00:23:25 --> 00:23:26
zeros there.
427
00:23:26 --> 00:23:29
Maybe the whole determinant is
zero.
428
00:23:29 --> 00:23:32
I mean, I -- is that a singular
matrix?
429
00:23:32 --> 00:23:35
That possibility definitely
exists.
430
00:23:35 --> 00:23:39
I could, I could,
So one way to do it would be
431
00:23:39 --> 00:23:40
elimination.
432
00:23:40 --> 00:23:44
Actually, that would probably
be a fairly reasonable way.
433
00:23:44 --> 00:23:47
I could use elimination,
so I could use -- go back to
434
00:23:47 --> 00:23:50
those properties,
that -- and use elimination,
435
00:23:50 --> 00:23:54
get down, eliminate it down,
do I have a row of zeros at the
436
00:23:54 --> 00:23:56.55
end of elimination?
437
00:23:56.55 --> 00:23:58
The answer is zero.
438
00:23:58 --> 00:24:01
I was thinking,
shall I try this big formula?
439
00:24:01 --> 00:24:02
OK.
440
00:24:02 --> 00:24:04
Let's try the big formula.
441
00:24:04 --> 00:24:08
How -- tell me one way I can go
down the matrix,
442
00:24:08 --> 00:24:11
taking a one,
taking a one from
443
00:24:11 --> 00:24:14
every row and column,
and make it to the end?
444
00:24:14 --> 00:24:17
So it's -- I get something that
isn't zero.
445
00:24:17 --> 00:24:21
Well, one way to do it,
I could take that times that
446
00:24:21 --> 00:24:23
times that times that times
that.
447
00:24:23 --> 00:24:26
That would be one and,
and, and I just said,
448
00:24:26 --> 00:24:29
that comes in with what sign?
449
00:24:29 --> 00:24:29
Plus.
450
00:24:29 --> 00:24:32
That comes with a plus sign.
451
00:24:32 --> 00:24:38
Because, because that
permutation -- I've just written
452
00:24:38 --> 00:24:44
the permutation about four three
two one, and one exchange and a
453
00:24:44 --> 00:24:49
second exchange,
two exchanges puts it in the
454
00:24:49 --> 00:24:50
correct order.
455
00:24:50 --> 00:24:53
Keep walking away,
don't....
456
00:24:53 --> 00:24:58
OK, we're executing a
determinant
457
00:24:58 --> 00:25:00
formula here.
458
00:25:00 --> 00:25:06
Uh as long as it's not
periodic, of course.
459
00:25:06 --> 00:25:11
If he comes back I'm in -- no.
460
00:25:11 --> 00:25:14
All right, all right.
461
00:25:14 --> 00:25:19
OK, so that would give me a
plus one.
462
00:25:19 --> 00:25:21
All right.
463
00:25:21 --> 00:25:24
Are there any others?
464
00:25:24 --> 00:25:30
Well, of course we see
another one here.
465
00:25:30 --> 00:25:35
This times this times this
times this strikes us right
466
00:25:35 --> 00:25:35
away.
467
00:25:35 --> 00:25:40.5
So that's the order three,
the order -- let me make a
468
00:25:40.5 --> 00:25:42
little different mark here.
469
00:25:42 --> 00:25:44
Three two one four.
470
00:25:44 --> 00:25:49
And is that a plus or a minus,
three two one four?
471
00:25:49 --> 00:25:53
Is that, is that permutation a
plus
472
00:25:53 --> 00:25:54.78
or a minus permutation?
473
00:25:54.78 --> 00:25:55
It's a minus.
474
00:25:55 --> 00:25:56
How do you see that?
475
00:25:56 --> 00:26:00
What exchange shall I do to get
it in the right order?
476
00:26:00 --> 00:26:04
If I exchange the one and the
three I'm in the right orders,
477
00:26:04 --> 00:26:08
took one exchange to do it,
so that would be a plus -- that
478
00:26:08 --> 00:26:10
would be a minus one.
479
00:26:10 --> 00:26:14
And now I don't know if
there're any more here.
480
00:26:14 --> 00:26:14
Let's see.
481
00:26:14 --> 00:26:17
Let me try again starting with
this.
482
00:26:17 --> 00:26:21
Now I've got to pick somebody
from -- oh yeah,
483
00:26:21 --> 00:26:24
see, you see what's happening.
484
00:26:24 --> 00:26:28
If I I start there,
OK, column three is used.
485
00:26:28 --> 00:26:31
So then when I go to next row,
I can't use that,
486
00:26:31 --> 00:26:31
I must use that.
487
00:26:31 --> 00:26:34
Now columns two and three are
used.
488
00:26:34 --> 00:26:36
When I come to this row I must
use that.
489
00:26:36 --> 00:26:37
And then I must use that.
490
00:26:37 --> 00:26:41
So if I start there,
this is the only one I get.
491
00:26:41 --> 00:26:44
And similarly,
if I start there,
492
00:26:44 --> 00:26:46.28
that's the only one I get.
493
00:26:46.28 --> 00:26:48
So what's the determinant?
494
00:26:48 --> 00:26:50
What's the determinant?
495
00:26:50 --> 00:26:50
Zero.
496
00:26:50 --> 00:26:53
The determinant is zero for
that case.
497
00:26:53 --> 00:26:57.76
Because we, we were able to
check the
498
00:26:57.76 --> 00:26:58
twenty-four terms.
499
00:26:58 --> 00:27:00.63
Twenty-two of them were zero.
500
00:27:00.63 --> 00:27:02
One of them was plus one.
501
00:27:02 --> 00:27:03
One of them was minus one.
502
00:27:03 --> 00:27:06
Add up the twenty-four terms,
zero is the answer.
503
00:27:06 --> 00:27:06
OK.
504
00:27:06 --> 00:27:10
Well, I didn't know it would be
zero, I -- because I wasn't,
505
00:27:10 --> 00:27:12
like, thinking ahead.
506
00:27:12 --> 00:27:15
I was a little scared,
actually.
507
00:27:15 --> 00:27:17
I said, that,
apparition went by.
508
00:27:17 --> 00:27:21
So and I don't know if the
camera caught that.
509
00:27:21 --> 00:27:26
So whether the rest of the
world will realize that I was in
510
00:27:26 --> 00:27:29
danger or not,
we don't know.
511
00:27:29 --> 00:27:33.82
But anyway, I guess he just
wanted to be sure
512
00:27:33.82 --> 00:27:37
that we got the right answer,
which is determinant zero.
513
00:27:37 --> 00:27:41
And then that makes me think,
OK, the matrix must be,
514
00:27:41 --> 00:27:43
the matrix must be singular.
515
00:27:43 --> 00:27:47
And then if the matrix is
singular, maybe there's another
516
00:27:47 --> 00:27:52
way to see that it's singular,
like find something in its null
517
00:27:52 --> 00:27:53
space.
518
00:27:53 --> 00:27:57
Or find a combination of the
rows that gives zero.
519
00:27:57 --> 00:28:02
And like what d- what,
what combination of those rows
520
00:28:02 --> 00:28:03
does give zero.
521
00:28:03 --> 00:28:06
Suppose I add rows one and rows
three.
522
00:28:06 --> 00:28:11
If I add rows one and rows
three, what do I get?
523
00:28:11 --> 00:28:14
I get a row of all ones.
524
00:28:14 --> 00:28:18
Then if I add rows two and rows
four I get a row of all ones.
525
00:28:18 --> 00:28:22
So row one minus row two plus
row three minus row four is
526
00:28:22 --> 00:28:23
probably the zero row.
527
00:28:23 --> 00:28:25
It's a singular matrix.
528
00:28:25 --> 00:28:28
And I could find something in
its null space the same way.
529
00:28:28 --> 00:28:33
That would be a combination of
columns that gives zero.
530
00:28:33 --> 00:28:35
OK, there's an example.
531
00:28:35 --> 00:28:35.93
All right.
532
00:28:35.93 --> 00:28:39
So that's, well,
that shows two things.
533
00:28:39 --> 00:28:43
That shows how we get the
twenty-four terms and it shows
534
00:28:43 --> 00:28:47
the great advantage of having a
lot of zeros in there.
535
00:28:47 --> 00:28:47.98
OK.
536
00:28:47.98 --> 00:28:54
So we'll use this big formula,
but I want to pick -- I want to
537
00:28:54 --> 00:28:57
go onward now to cofactors.
538
00:28:57 --> 00:28:59
Onward to cofactors.
539
00:28:59 --> 00:29:04.62
Cofactors is a way of breaking
up this big formula that
540
00:29:04.62 --> 00:29:11
connects this n by n -- this is
an n by n determinant that we've
541
00:29:11 --> 00:29:15.08
just have a formula for,
the big formula.
542
00:29:15.08 --> 00:29:20
So cofactors is a way to
connect this n by n determinant
543
00:29:20 --> 00:29:24
to,
determinants one smaller.
544
00:29:24 --> 00:29:25
One smaller.
545
00:29:25 --> 00:29:32
And the way we want to do it is
actually going to show up in
546
00:29:32 --> 00:29:32
this.
547
00:29:32 --> 00:29:38
Since the three by three is the
one that we wrote out in full,
548
00:29:38 --> 00:29:44
let's, let me do this three by
-- so I'm talking about
549
00:29:44 --> 00:29:51
cofactors, and I'm going to
start again with three by three.
550
00:29:51 --> 00:29:55
And I'm going to take the,
the exact formula,
551
00:29:55 --> 00:30:00
and I'm just going to write it
as a one one -- this is the
552
00:30:00 --> 00:30:02
determinant I'm writing.
553
00:30:02 --> 00:30:06
I'm just going to say a one one
times what?
554
00:30:06 --> 00:30:08
A one one times what?
555
00:30:08 --> 00:30:14
And it's a one one times a two
two a three three minus a two
556
00:30:14 --> 00:30:16
three a three two.
557
00:30:16 --> 00:30:22
Then I've got the a one two
stuff times something.
558
00:30:22 --> 00:30:28
And I've got the a one three
stuff times something.
559
00:30:28 --> 00:30:30
Do you see what I'm doing?
560
00:30:30 --> 00:30:36
I'm taking our big formula and
I'm saying, OK,
561
00:30:36 --> 00:30:41
choose column -- out of the
first row,
562
00:30:41 --> 00:30:43
choose column one.
563
00:30:43 --> 00:30:46
And take all the possibilities.
564
00:30:46 --> 00:30:53
And those extra factors will be
what we'll call the cofactor,
565
00:30:53 --> 00:30:56
co meaning going with a one
one.
566
00:30:56 --> 00:31:00
So this in parenthesis are,
these are in,
567
00:31:00 --> 00:31:03.81
the cofactors are in parens.
568
00:31:03.81 --> 00:31:06
A one one times something.
569
00:31:06 --> 00:31:12
And I figured out what that
something was by just looking
570
00:31:12 --> 00:31:17
back -- if I can walk back here
to the, to the a one one,
571
00:31:17 --> 00:31:21
the one that comes down the
diagonal minus the one that
572
00:31:21 --> 00:31:22
comes that way.
573
00:31:22 --> 00:31:26.76
That's, those are the two,
only two that used a one one.
574
00:31:26.76 --> 00:31:30
So there they are,
one with a plus and one with a
575
00:31:30 --> 00:31:30
minus.
576
00:31:30 --> 00:31:36
And now I can write in the --
I could look back and see what
577
00:31:36 --> 00:31:40
used a one two and I can see
what used a one three,
578
00:31:40 --> 00:31:45
and those will give me the
cofactors of a one two and a one
579
00:31:45 --> 00:31:45
three.
580
00:31:45 --> 00:31:48
Before I do that,
what's this number,
581
00:31:48 --> 00:31:50
what is this cofactor?
582
00:31:50 --> 00:31:55.13
What is it there that's
multiplying a one one?
583
00:31:55.13 --> 00:32:01
Tell me what a two two a three
three minus a two three a three
584
00:32:01 --> 00:32:05.57
two is, for this -- do you
recognize that?
585
00:32:05.57 --> 00:32:11
Do you recognize -- let's see,
I can -- and I'll put it here.
586
00:32:11 --> 00:32:13
There's the a one one.
587
00:32:13 --> 00:32:16
That's used column one.
588
00:32:16 --> 00:32:21.64
Then there's --
the other factors involved
589
00:32:21.64 --> 00:32:23
these other columns.
590
00:32:23 --> 00:32:25
This row is used.
591
00:32:25 --> 00:32:27
This column is used.
592
00:32:27 --> 00:32:32
So this the only things left to
use are these.
593
00:32:32 --> 00:32:37
And this formula uses them,
and what's the,
594
00:32:37 --> 00:32:39
what's the cofactor?
595
00:32:39 --> 00:32:46.33
Tell me what it is because you
see it, and then --
596
00:32:46.33 --> 00:32:50
I'll be happy you see what the
idea of cofactors.
597
00:32:50 --> 00:32:54.39
It's the determinant of this
smaller guy.
598
00:32:54.39 --> 00:32:59
A one one multiplies the
determinant of this smaller guy.
599
00:32:59 --> 00:33:04
That gives me all the a one one
part of the big formula.
600
00:33:04 --> 00:33:05
You see that?
601
00:33:05 --> 00:33:11
This, the determinant of this
smaller guy is a two two a
602
00:33:11 --> 00:33:15
three three minus a two three a
three two.
603
00:33:15 --> 00:33:19
In other words,
once I've used column one and
604
00:33:19 --> 00:33:24
row one, what's left is all the
ways to use the other n-1
605
00:33:24 --> 00:33:26
columns and n-1 rows,
one of each.
606
00:33:26 --> 00:33:32
All the other -- and that's the
determinant of the smaller guy
607
00:33:32 --> 00:33:34
of size n-1.
608
00:33:34 --> 00:33:37
So that's the whole idea of
cofactors.
609
00:33:37 --> 00:33:42
And we just have to remember
that with determinants we've got
610
00:33:42 --> 00:33:46
pluses and minus signs to keep
straight.
611
00:33:46 --> 00:33:49
Can we keep this next one
straight?
612
00:33:49 --> 00:33:51
Let's do the next one.
613
00:33:51 --> 00:33:54
OK, the next one will be when I
use a one two.
614
00:33:54 --> 00:34:00
I'll have left --
so I can't use that column any
615
00:34:00 --> 00:34:07
more, but I can use a two one
and a two three and I can use a
616
00:34:07 --> 00:34:10
three one and a three three.
617
00:34:10 --> 00:34:15
So this one gave me a one times
that determinant.
618
00:34:15 --> 00:34:20
This will give me a one two
times this determinant,
619
00:34:20 --> 00:34:27
a two one a three three minus a
two three a three one.
620
00:34:27 --> 00:34:30
So that's all the stuff
involving a one two.
621
00:34:30 --> 00:34:33.46
But have I got the sign right?
622
00:34:33.46 --> 00:34:38
Is the determinant of that
correctly given by that or is
623
00:34:38 --> 00:34:39
there a minus sign?
624
00:34:39 --> 00:34:41.54
There is a minus sign.
625
00:34:41.54 --> 00:34:43
I can follow one of these.
626
00:34:43 --> 00:34:48
If I do that times that times
that, that was one that's
627
00:34:48 --> 00:34:52
showing up
here, but it should have showed
628
00:34:52 --> 00:34:55
-- it should have been a minus.
629
00:34:55 --> 00:34:59.91
So I'm going to build that
minus sign into the cofactor.
630
00:34:59.91 --> 00:35:04
So, so the cofactor -- so I'll
put, put that minus sign in
631
00:35:04 --> 00:35:05.14
here.
632
00:35:05.14 --> 00:35:09
So because the cofactor is
going to be strictly the thing
633
00:35:09 --> 00:35:13
that multiplies the,
the factor.
634
00:35:13 --> 00:35:16
The factor is a one two,
the cofactor is this,
635
00:35:16 --> 00:35:19.43
is the parens,
the stuff in parentheses.
636
00:35:19.43 --> 00:35:21
So it's got the minus sign
built in.
637
00:35:21 --> 00:35:25
And if I did -- if I went on to
the third guy,
638
00:35:25 --> 00:35:28
there w- there'll be this and
this, this and this.
639
00:35:28 --> 00:35:31
And it would take its
determinant.
640
00:35:31 --> 00:35:34
It would come out plus
the determinant.
641
00:35:34 --> 00:35:38
So now I'm ready to say what
cofactors are.
642
00:35:38 --> 00:35:42
So this would be a plus and a
one three times its cofactor.
643
00:35:42 --> 00:35:47
And over here we had plus a one
one times this determinant.
644
00:35:47 --> 00:35:51
But and there we had the a one
two times its cofactor,
645
00:35:51 --> 00:35:58
but the -- so the point is the
cofactor is either plus or
646
00:35:58 --> 00:36:00
minus the determinant.
647
00:36:00 --> 00:36:05
So let me write that underneath
them.
648
00:36:05 --> 00:36:09
What is the,
what are cofactors?
649
00:36:09 --> 00:36:14
The cofactor if any number aij,
let's say.
650
00:36:14 --> 00:36:19
This is, this is all the terms
in the,
651
00:36:19 --> 00:36:23
in the big formula that involve
aij.
652
00:36:23 --> 00:36:29
We're especially interested in
a1j, the first row,
653
00:36:29 --> 00:36:35
that's what I've been talking
about, but any row would be all
654
00:36:35 --> 00:36:36
right.
655
00:36:36 --> 00:36:40
All right, so -- what terms
involve aij?
656
00:36:40 --> 00:36:45
So --
it's the determinant of the n
657
00:36:45 --> 00:36:51
minus one matrix -- with row i,
column j erased.
658
00:36:51 --> 00:36:57.23
So it's the,
it's a matrix of size n-1 with
659
00:36:57.23 --> 00:37:02.86
-- of course,
because I can't use this row or
660
00:37:02.86 --> 00:37:05
this column again.
661
00:37:05 --> 00:37:09
So I have the matrix all there.
662
00:37:09 --> 00:37:15
But now it's multiplied
by a plus or a minus.
663
00:37:15 --> 00:37:19
This is the cofactor,
and I'm going to call that cij.
664
00:37:19 --> 00:37:25
Capital, I use capital c just
to, just to emphasize that these
665
00:37:25 --> 00:37:29
are important and emphasize that
they're, they're,
666
00:37:29 --> 00:37:32
they're different from the
(a)s.
667
00:37:32 --> 00:37:32
OK.
668
00:37:32 --> 00:37:36
So now is it a plus
or is it a minus?
669
00:37:36 --> 00:37:42
Because we see that in this
case, for a one one it was a
670
00:37:42 --> 00:37:46
plus, for a one two I -- this is
ij -- it was a minus.
671
00:37:46 --> 00:37:49
For this ij it was a plus.
672
00:37:49 --> 00:37:54
So any any guess on the rule
for plus or minus when we see
673
00:37:54 --> 00:37:58
those examples,
ij equal one one or one three
674
00:37:58 --> 00:38:00
was a plus?
675
00:38:00 --> 00:38:05
It sounds very like i+j odd or
even.
676
00:38:05 --> 00:38:12
That, that's doesn't surprise
us, and that's the right answer.
677
00:38:12 --> 00:38:20
So it's a plus if i+j is even
and it's a minus if i+j is odd.
678
00:38:20 --> 00:38:26
So if I go along row one and
look at the cofactors,
679
00:38:26 --> 00:38:34
I just take those determinants,
those one smaller determinants,
680
00:38:34 --> 00:38:38
and
they come in order plus minus
681
00:38:38 --> 00:38:40
plus minus plus minus.
682
00:38:40 --> 00:38:45
But if I go along row two and,
and, and take the cofactors of
683
00:38:45 --> 00:38:49
sub-determinants,
they would start with a minus,
684
00:38:49 --> 00:38:53
because the two one entry,
two plus one is odd,
685
00:38:53 --> 00:38:58
so the -- like there's a
pattern plus minus plus minus
686
00:38:58 --> 00:39:05
plus if it was five by
five, but then if I was doing a
687
00:39:05 --> 00:39:12
cofactor then this sign would be
minus plus minus plus minus,
688
00:39:12 --> 00:39:17
plus minus plus -- it's sort of
checkerboard.
689
00:39:17 --> 00:39:17
OK.
690
00:39:17 --> 00:39:17
OK.
691
00:39:17 --> 00:39:24.13
Those are the signs that,
that are given by this rule,
692
00:39:24.13 --> 00:39:25
i+j even or odd.
693
00:39:25 --> 00:39:31
And those are built
into the cofactors.
694
00:39:31 --> 00:39:36
The thing is called a minor
without th- before you've built
695
00:39:36 --> 00:39:40
in the sign, but I don't care
about those.
696
00:39:40 --> 00:39:44.02
Build in that sign and call it
a cofactor.
697
00:39:44.02 --> 00:39:44
OK.
698
00:39:44 --> 00:39:47
So what's the cofactor formula?
699
00:39:47 --> 00:39:50
What's the cofactor formula
then?
700
00:39:50 --> 00:39:54
Let me come back to this board
and
701
00:39:54 --> 00:39:58
say, what's the cofactor
formula?
702
00:39:58 --> 00:40:04
Determinant of A is -- let's go
along the first row.
703
00:40:04 --> 00:40:11
It's a one one times its
cofactor, and then the second
704
00:40:11 --> 00:40:18
guy is a one two times its
cofactor, and you just keep
705
00:40:18 --> 00:40:25
going to the end of the row,
a1n times its cofactor.
706
00:40:25 --> 00:40:31
So that's cofactor for --
along row one.
707
00:40:31 --> 00:40:38
And if I went along row I,
I would -- those ones would be
708
00:40:38 --> 00:40:38
Is.
709
00:40:38 --> 00:40:42
That's worth putting a box
over.
710
00:40:42 --> 00:40:46
That's the cofactor formula.
711
00:40:46 --> 00:40:53
Do you see that -- actually,
this would give me another way
712
00:40:53 --> 00:40:59
I could have started the whole
topic of determinants.
713
00:40:59 --> 00:41:05
And some, some people might do
it
714
00:41:05 --> 00:41:07
this -- choose to do it this
way.
715
00:41:07 --> 00:41:11
Because the cofactor formula
would allow me to build up an n
716
00:41:11 --> 00:41:14
by n determinant out of n-1
sized determinants,
717
00:41:14 --> 00:41:17
build those out of n-2,
and so on.
718
00:41:17 --> 00:41:19.99
I could boil all the way down
to one by ones.
719
00:41:19.99 --> 00:41:23
So what's the cofactor formula
for
720
00:41:23 --> 00:41:24
two by two matrices?
721
00:41:24 --> 00:41:26
Yeah, tell me that.
722
00:41:26 --> 00:41:28
What's the cofactor for us?
723
00:41:28 --> 00:41:32
Here is the,
here is the world's smallest
724
00:41:32 --> 00:41:35
example, practically,
of a cofactor formula.
725
00:41:35 --> 00:41:36
OK.
726
00:41:36 --> 00:41:38
Let's go along row one.
727
00:41:38 --> 00:41:42
I take this first guy
times its cofactor.
728
00:41:42 --> 00:41:46
What's the cofactor of the one
one entry?
729
00:41:46 --> 00:41:51
d, because you strike out the
one one row and column and
730
00:41:51 --> 00:41:53
you're left with d.
731
00:41:53 --> 00:41:57
Then I take this guy,
b, times its cofactor.
732
00:41:57 --> 00:42:00
What's the cofactor of b?
733
00:42:00 --> 00:42:05
Is it c or it's --
minus c, because I strike out
734
00:42:05 --> 00:42:10
this guy, I take that
determinant, and then I follow
735
00:42:10 --> 00:42:15
the i+j rule and I get a minus,
I get an odd.
736
00:42:15 --> 00:42:17
So it's b times minus c.
737
00:42:17 --> 00:42:19
OK, it worked.
738
00:42:19 --> 00:42:21
Of course it,
it worked.
739
00:42:21 --> 00:42:24
And the three by three works.
740
00:42:24 --> 00:42:29
So that's the cofactor formula,
and that is,
741
00:42:29 --> 00:42:33
that's an -- that's a good
formula to know,
742
00:42:33 --> 00:42:39
and now I'm feeling like,
wow, I'm giving you a lot of
743
00:42:39 --> 00:42:41
algebra to swallow here.
744
00:42:41 --> 00:42:45
Last lecture gave you ten
properties.
745
00:42:45 --> 00:42:51
Now I'm giving you -- and by
the way, those ten
746
00:42:51 --> 00:42:55
properties led us to a formula
for the determinant which was
747
00:42:55 --> 00:42:58
very important,
and I haven't repeated it till
748
00:42:58 --> 00:42:59
now.
749
00:42:59 --> 00:43:00
What was that?
750
00:43:00 --> 00:43:03
The, the determinant is the
product of the pivots.
751
00:43:03 --> 00:43:06
So the pivot formula is,
is very important.
752
00:43:06 --> 00:43:09
The pivots have all this
complicated
753
00:43:09 --> 00:43:11
mess already built in.
754
00:43:11 --> 00:43:16
As you did elimination to get
the pivots, you built in all
755
00:43:16 --> 00:43:19
this horrible stuff,
quite efficiently.
756
00:43:19 --> 00:43:23
Then the big formula with the n
factorial terms,
757
00:43:23 --> 00:43:27
that's got all the horrible
stuff spread out.
758
00:43:27 --> 00:43:32
And the cofactor formula is
like in between.
759
00:43:32 --> 00:43:36
It's got easy stuff times
horrible stuff,
760
00:43:36 --> 00:43:37
basically.
761
00:43:37 --> 00:43:42
But it's, it shows you,
how to get determinants from
762
00:43:42 --> 00:43:47
smaller determinants,
and that's the application that
763
00:43:47 --> 00:43:49
I now want to make.
764
00:43:49 --> 00:43:52.7
So may I do one more example?
765
00:43:52.7 --> 00:43:55
So I remember the general idea.
766
00:43:55 --> 00:44:00
But I'm going to use this
cofactor
767
00:44:00 --> 00:44:06
formula for a matrix -- so here
is going to be my example.
768
00:44:06 --> 00:44:11.4
It's -- I promised in the,
in the lecture,
769
00:44:11.4 --> 00:44:16
outline at the very beginning
to do an example.
770
00:44:16 --> 00:44:22
And let me do -- I'm going to
pick tri-diagonal matrix of
771
00:44:22 --> 00:44:24
ones.
772
00:44:24 --> 00:44:30.01
I could, I'm drawing here the
four by four.
773
00:44:30.01 --> 00:44:33
So this will be the matrix.
774
00:44:33 --> 00:44:36
I could call that A4.
775
00:44:36 --> 00:44:40.93
But my real idea is to do n by
n.
776
00:44:40.93 --> 00:44:42
To do them all.
777
00:44:42 --> 00:44:51
So A -- I could -- everybody
understands what A1 and A2 are.
778
00:44:51 --> 00:44:53
Yeah.
779
00:44:53 --> 00:44:58.63
Maybe we should just do A1 and
A2 and A3 just for -- so this is
780
00:44:58.63 --> 00:44:58
A4.
781
00:44:58 --> 00:45:01
What's the determinant of A1?
782
00:45:01 --> 00:45:04.09
What's the determinant of A1?
783
00:45:04.09 --> 00:45:07
So, so what's the matrix A1 in
this formula?
784
00:45:07 --> 00:45:09.64
It's just got that.
785
00:45:09.64 --> 00:45:11
So the determinant is one.
786
00:45:11 --> 00:45:14
What's the determinant of A2?
787
00:45:14 --> 00:45:18
So it's just got this two by
two,
788
00:45:18 --> 00:45:20
and its determinant is -- zero.
789
00:45:20 --> 00:45:23
And then the three by three.
790
00:45:23 --> 00:45:25
Can we see its determinant?
791
00:45:25 --> 00:45:29
Can you take the determinant of
that three by three?
792
00:45:29 --> 00:45:33
Well, that's not quite so
obvious, at least not to me.
793
00:45:33 --> 00:45:37
Being three by three,
I don't know -- so here's a,
794
00:45:37 --> 00:45:40
here's a good example.
795
00:45:40 --> 00:45:43
How would you do that three by
three determinant?
796
00:45:43 --> 00:45:46
We've got, like,
n factorial different ways.
797
00:45:46 --> 00:45:47
Well, three factorial.
798
00:45:47 --> 00:45:49
So we've got six ways.
799
00:45:49 --> 00:45:49
OK.
800
00:45:49 --> 00:45:53
I mean, one way to do it --
actually the way I would
801
00:45:53 --> 00:45:55
probably do it,
being three by three,
802
00:45:55 --> 00:45:58
I would use the complete the
big formula.
803
00:45:58 --> 00:46:00
I would say,
I've got a one from that,
804
00:46:00 --> 00:46:03
I've got a zero from that,
I've got a zero from that,
805
00:46:03 --> 00:46:05
a zero from that,
and this direction is a minus
806
00:46:05 --> 00:46:09.01
one,
that direction's a minus one.
807
00:46:09.01 --> 00:46:11
I believe the answer is minus
one.
808
00:46:11 --> 00:46:13
Would you do it another way?
809
00:46:13 --> 00:46:16
Here's another way to do it,
look.
810
00:46:16 --> 00:46:21
Subtract row three from -- I'm
just looking at this three by
811
00:46:21 --> 00:46:21
three.
812
00:46:21 --> 00:46:25
Everybody's looking at the
three by three.
813
00:46:25 --> 00:46:28
Subtract row three from row
two.
814
00:46:28 --> 00:46:30
Determinant doesn't change.
815
00:46:30 --> 00:46:32.44
So those become zeros.
816
00:46:32.44 --> 00:46:34
OK, now use the cofactor
formula.
817
00:46:34 --> 00:46:35
How's that?
818
00:46:35 --> 00:46:40
How can, how -- if this was now
zeros and I'm looking at this
819
00:46:40 --> 00:46:43
three by three,
use the cofactor formula.
820
00:46:43 --> 00:46:48
Why not use the cofactor
formula along that row?
821
00:46:48 --> 00:46:53
Because then I take that number
times its cofactor,
822
00:46:53 --> 00:46:59
so I take this number -- let me
put a box around it -- times its
823
00:46:59 --> 00:47:04.97
cofactor, which is the
determinant of that and that,
824
00:47:04.97 --> 00:47:06
which is what?
825
00:47:06 --> 00:47:10
That two by two matrix has
determinant one.
826
00:47:10 --> 00:47:13
So what's the cofactor?
827
00:47:13 --> 00:47:17
What's the cofactor of this guy
here?
828
00:47:17 --> 00:47:21
Looking just at this three by
three.
829
00:47:21 --> 00:47:25
The cofactor of that one is
this determinant,
830
00:47:25 --> 00:47:28
which is one times negative.
831
00:47:28 --> 00:47:33
So that's why the answer came
out minus one.
832
00:47:33 --> 00:47:34.59
OK.
833
00:47:34.59 --> 00:47:36
So I did the three by three.
834
00:47:36 --> 00:47:39
I don't know if we want to try
the four by four.
835
00:47:39 --> 00:47:42
Yeah, let's -- I guess that was
the point of my example,
836
00:47:42 --> 00:47:44
of course, so I have to try it.
837
00:47:44 --> 00:47:47
Sorry, I'm in a good mood
today, so you have to stand for
838
00:47:47 --> 00:47:49.59
all the bad jokes.
839
00:47:49.59 --> 00:47:49
OK.
840
00:47:49 --> 00:47:50
OK.
841
00:47:50 --> 00:47:52.95
So what was the matrix?
842
00:47:52.95 --> 00:47:53
Ah.
843
00:47:53 --> 00:47:57
OK, now I'm ready for four by
four.
844
00:47:57 --> 00:48:04
Who wants to -- who wants to
guess the, the -- I don't know,
845
00:48:04 --> 00:48:10
frankly, this four by four,
what's, what's the determinant.
846
00:48:10 --> 00:48:14
I plan to use cofactors.
847
00:48:14 --> 00:48:16
OK, let's use cofactors.
848
00:48:16 --> 00:48:21
The determinant of A4 is -- OK,
let's use cofactors on the
849
00:48:21 --> 00:48:22
first row.
850
00:48:22 --> 00:48:23.77
Those are easy.
851
00:48:23.77 --> 00:48:28
So I multiply this number,
which is a convenient one,
852
00:48:28 --> 00:48:30
times this determinant.
853
00:48:30 --> 00:48:34
So it's, it's one times the,
this three by three
854
00:48:34 --> 00:48:35
determinant.
855
00:48:35 --> 00:48:40
Now what is -- do you recognize
that matrix?
856
00:48:40 --> 00:48:40
It's A3.
857
00:48:40 --> 00:48:44
So it's one times the
determinant of A3.
858
00:48:44 --> 00:48:49
Coming along this row is a one
times this determinant,
859
00:48:49 --> 00:48:52
and it goes with a plus,
right?
860
00:48:52 --> 00:48:55
And then we have this one.
861
00:48:55 --> 00:48:58
And what is its cofactor?
862
00:48:58 --> 00:49:01
Now I'm looking at,
now I'm looking at this three
863
00:49:01 --> 00:49:05
by three, this three by three,
so I'm looking at the three by
864
00:49:05 --> 00:49:07
three that I haven't X-ed out.
865
00:49:07 --> 00:49:11.12
What is that -- oh,
now it, we did a plus or a --
866
00:49:11.12 --> 00:49:14
is it plus this determinant,
this three by three
867
00:49:14 --> 00:49:16
determinant, or minus it?
868
00:49:16 --> 00:49:19
It's minus it,
right, because this is -- I'm
869
00:49:19 --> 00:49:23
starting in a one two position,
and that's a minus.
870
00:49:23 --> 00:49:26
So I want minus this
determinant.
871
00:49:26 --> 00:49:28
But these guys are X-ed out.
872
00:49:28 --> 00:49:28
OK.
873
00:49:28 --> 00:49:30
So I've got a three by three.
874
00:49:30 --> 00:49:32
Well, let's use cofactors
again.
875
00:49:32 --> 00:49:38
Use cofactors of the column,
because actually we could use
876
00:49:38 --> 00:49:41
cofactors of columns just as
well as rows,
877
00:49:41 --> 00:49:44.64
because, because the transpose
rule.
878
00:49:44.64 --> 00:49:48
So I'll take this one,
which is now sitting in the
879
00:49:48 --> 00:49:52
plus position,
times its determinant -- oh!
880
00:49:52 --> 00:49:52
Oh, hell.
881
00:49:52 --> 00:49:57
What -- oh yeah,
I shouldn't have said hell,
882
00:49:57 --> 00:49:58
because it's all right.
883
00:49:58 --> 00:49:58
OK.
884
00:49:58 --> 00:50:00
One times the determinant.
885
00:50:00 --> 00:50:03
What is that matrix now that
I'm taking the,
886
00:50:03 --> 00:50:04
this smaller one of?
887
00:50:04 --> 00:50:06
Oh, but there's a minus,
right?
888
00:50:06 --> 00:50:10.4
The minus came from,
from the fact that this was in
889
00:50:10.4 --> 00:50:13
the one two position and that's
odd.
890
00:50:13 --> 00:50:20
So this is a minus one times --
and what's -- and then this one
891
00:50:20 --> 00:50:25
is the upper left,
that's the one one position in
892
00:50:25 --> 00:50:27
its matrix, so plus.
893
00:50:27 --> 00:50:30
And what's this matrix here?
894
00:50:30 --> 00:50:32
Do you recognize that?
895
00:50:32 --> 00:50:36
That matrix is -- yes,
please say it -- A2.
896
00:50:36 --> 00:50:41
And we -- that's our formula
for any case.
897
00:50:41 --> 00:50:47.81
A of any size n is equal to the
determinant of A n minus one,
898
00:50:47.81 --> 00:50:53
that's what came from taking
the one in the upper corner,
899
00:50:53 --> 00:50:57
the first cofactor,
minus the determinant of A n
900
00:50:57 --> 00:50:58
minus two.
901
00:50:58 --> 00:51:02
What we discovered there is
true for all n.
902
00:51:02 --> 00:51:08
I didn't even mention it,
but I stopped taking cofactors
903
00:51:08 --> 00:51:10
when I got this one.
904
00:51:10 --> 00:51:11
Why did I stop?
905
00:51:11 --> 00:51:15
Why didn't I take the cofactor
of this guy?
906
00:51:15 --> 00:51:19
Because he's going to get
multiplied by zero,
907
00:51:19 --> 00:51:21
and no, no use wasting time.
908
00:51:21 --> 00:51:22
Or this one too.
909
00:51:22 --> 00:51:27
The cofactor,
her cofactor will be whatever
910
00:51:27 --> 00:51:30
that determinant is,
but it'll be multiplied by
911
00:51:30 --> 00:51:32
zero, so I won't bother.
912
00:51:32 --> 00:51:34
OK, there is the formula.
913
00:51:34 --> 00:51:39
And that now tells us -- I
could figure out what A4 is now.
914
00:51:39 --> 00:51:42
Oh yeah, finally I can get A4.
915
00:51:42 --> 00:51:45
Because it's A3,
which is minus one,
916
00:51:45 --> 00:51:49
minus A2, which is zero,
so it's minus one.
917
00:51:49 --> 00:51:54
You see how we're getting kind
of numbers that you might not
918
00:51:54 --> 00:51:55
have guessed.
919
00:51:55 --> 00:52:00
So our sequence right now is
one zero minus one minus one.
920
00:52:00 --> 00:52:03.71
What's the next one in the
sequence, A5?
921
00:52:03.71 --> 00:52:08
A5 is this minus this,
so it is zero.
922
00:52:08 --> 00:52:09
What's A6?
923
00:52:09 --> 00:52:13
A6 is this minus this,
which is one.
924
00:52:13 --> 00:52:14
What's A7?
925
00:52:14 --> 00:52:22
I'm, I'm going to be stopped by
either the time runs out or the
926
00:52:22 --> 00:52:24
board runs out.
927
00:52:24 --> 00:52:28
OK, A7 is this minus this,
which is one.
928
00:52:28 --> 00:52:33
I'll stop here,
because time is out,
929
00:52:33 --> 00:52:36.61
but let me tell you what we've
got.
930
00:52:36.61 --> 00:52:41
What -- these determinants have
this series, one zero minus one
931
00:52:41 --> 00:52:45
minus one zero one,
and then it starts repeating.
932
00:52:45 --> 00:52:47
It's pretty fantastic.
933
00:52:47 --> 00:52:50.41
These determinants have period
six.
934
00:52:50.41 --> 00:52:54
So the determinant of A
sixty-one
935
00:52:54 --> 00:52:58
would be the determinant of A1,
which would be one.
936
00:52:58 --> 00:52:58
OK.
937
00:52:58 --> 00:53:01
I hope you liked that example.
938
00:53:01 --> 00:53:05
A non-trivial example of a
tri-diagonal determinant.
939
00:53:05 --> 00:53:05
Thanks.
940
00:53:05 --> 00:53:08
See you on Wednesday.