OK, this is the second
lecture on determinants. There are only three. With determinants it's a
fascinating, small topic inside linear algebra. Used to be determinants
were the big thing, and linear algebra was the
little thing, but they -- those changed, that
situation changed. Now determinants is one specific
part, very neat little part. And my goal today is to find
a formula for the determinant. It'll be a messy formula. So that's why you didn't
see it right away. But if I'm given
this n by n matrix then I use those
entries to create this number, the determinant. So there's a formula for it. In fact, there's another
formula, a second formula using something called cofactors. So you'll -- you have to
know what cofactors are. And then I'll apply
those formulas for some, some matrices
that have a lot of zeros away from the three diagonals. OK. So I'm shooting now for a
formula for the determinant. You remember we started with
these three properties, three simple properties,
but out of that we got all these amazing facts,
like the determinant of A B equals determinant of A
times determinant of B. But the three facts were -- oh, how about I just
take two by twos. I know, because everybody here
knows, the determinant of a two by two matrix, but let's get
it out of these three formulas. OK, so here's my, my
two by two matrix. I'm looking for a formula
for this determinant. a b c d, OK. So property one, I know what
to do with the identity. Right? Property two allows
me to exchange rows, and I know what to do then. So I know that that
determinant is one. Property two allows me
to exchange rows and know that this determinant
is minus one. And now I want to use property
three to get everybody, to get everybody. And how will I do that? OK. So remember that if I keep
the second row the same, I'm allowed to use
linearity in the first row. And I'll just use
it in a simple way. I'll write this vector
a b as a 0 + 0 b. So that's one step using
property three, linearity in the first row when the
second row's the same. OK. But now you can guess
what I'm going to do next. I'll -- because I'd like to -- if I can make the
matrices diagonal, then I'm clearly there. So I'll take this one. Now I'll keep the first row
fixed and split the second row, so that'll be an a 0
and I'll split that into a c 0 and, keeping that
first row the same, a 0 d. I used, for this
part, linearity. And now I'll -- whoops, that's
plus because I've got more coming. This one I'll do the same. I'll keep this
first row the same and I'll split c d
into c 0 and 0 d. OK. Now I've got four
easy determinants, and two of them are -- well, all four are
extremely easy. Two of them are so easy as
to turn into zero, right? Which two of these determinants
are zero right away? The first guy is zero. Why is he zero? Why is that determinant
nothing, forget him? Well, it has a column of zeros. And by the -- well, so
one way to think is, well, it's a singular matrix. Oh, for, for like forty-eight
different reasons, that determinant is zero. It's a singular matrix
that has a column of zeros. It's, it's dead. And this one is
about as dead too. Column of zeros. OK. So that's leaving
us with this one. Now what do I -- how do
I know its determinant, following the rules? Well, I guess one of the
properties that we actually got to was the determinant of that
-- diagonal matrix, then -- so I, I'm finally getting to
that determinant is the a d. And this determinant is what? What's this one? Minus -- because I would use
property two to do a flip to make it c b, then property
three to factor out the b, property c to
factor out the c -- the property again to factor
out the c, and that minus, and of course finally I got the
answer that we knew we would get. But you see the method. You see the method, because it's
method I'm looking for here, not just a two by two answer
but the method of doing -- now I can do three by threes
and four by fours and any size. So if you can see the method
of taking each row at a time -- so let's -- what would
happen with three by threes? Can we mentally do
it rather than I write everything on the
board for three by threes? So what would we do if
I had three by threes? I would keep rows two
and three the same and I would split the first
row into how many pieces? Three pieces. I'd have an A zero
zero and a zero B zero and a zero zero C or
something for the first row. So I would instead of going from
one piece to two pieces to four pieces, I would go from one
piece to three pieces to -- what would it be? Each of those three,
would, would it be nine? Or twenty-seven? Oh yeah, I've actually
got more steps, right. I'd go to nine but then I'd have
another row to straighten out, twenty-seven. Yes, oh God. OK, let me say this again then. If I -- if it was three
by three, I would -- separating out one
row into three pieces would give me three, separating
out the second row into three pieces, then I'd be up to nine,
separating out the third row into its three pieces, I'd be
up to twenty-seven, three cubed, pieces. But a lot of them would be zero. So now when would
they not be zero? Tell me the pieces
that would not be zero. Now I will write
the non-zero ones. OK, so I have this matrix. I think I have to
use these, start using these double symbols here
because otherwise I could never do n by n. OK. OK. So I split this up like crazy. A bunch of pieces are zero. Whenever I have a column of
zeros, I know I've got zero. When do I not have zero? When do I have -- what is
it that's like these guys? These are the survivors,
two survivors there. So my question
for three by three is going to be what
are the survivors? How many survivors are there? What are they? And when do I get a survivor. Well, I would get a survivor -- for example, one
survivor will be that one times that one times
that one, with all zeros everywhere else. That would be one survivor. a one one zero zero
zero a two two zero zero zero a three three. That's like the a d survivor. Tell me another survivor. What other thing -- oh,
now here you see the clue. Now can -- shall I just
say the whole clue? That I'm having -- the survivors have one entry
from each row and each column. One entry from each
row and column. Because if some
column is missing, then I get a singular matrix. And that, that's
one of these guys. See, you see what
happened with -- this guy? Column one never
got used in 0 b 0 d. So its determinant was
zero and I forget it. So I'm going to forget
those and just put -- so tell me one more that
would be a survivor? Well -- well,
here's another one. a one one zero zero -- now OK,
that's used up row -- row one is used. Column one is already
used so it better be zero. What else could I have? Where could I pick the guy --
which column shall I use in row two? Use column three, because
here if I use column -- here I used column
one and row one. This was like the column -- numbers were one two
three, right in order. Now the column numbers are going
to be one three, column three, and column two. So the row numbers are
one two three, of course. The column numbers are some -- OK, some permutation
of one two three, and here they come in
the order one three two. It's just like having
a permutation matrix with, instead of the
ones, with numbers. And actually, it's very close
to having a permutation matrix, because I, what I do eventually
is I factor out these numbers and then I have got. So what is that
determinant equal? I factor those
numbers out and I've got a one one times a two
two times a three three. And what does this
determinant equal? Yeah, now tell me the, this -- I mean, we're really getting
to the heart of these formulas now. What is that determinant? By the laws of -- by,
by our three properties, I can factor these out, I
can factor out the a one one, the a two three,
and the a three two. They're in separate rows. I can do each row separately. And then I just have to
decide is that plus sign or is that a minus sign? And the answer is it's a minus. Why minus? Because these is
one row exchange to get it back to the identity. So that's a minus. Now I through? No, because there
are other ways. What I'm really
through with, what I've done, what I've,
what I've completed is only the part where
the a one one is there. But now I've got parts
where it's a one two. And now if it's a one two that
row is used, that column is used. You see that idea? I could use this row and column. Now that column is used,
that column is used, and this guy has to be
here, a three three. And what's that determinant? That's an a one two times an a
two one times an a three three, and does it have
a plus or a minus? A minus is right. It has a minus. Because it's one
flip away from an id- from the, regular, the right
order, the diagonal order. And now what's the other
guy with a -- with, a one two up there? I could have used this row. I could have put this guy
here and this guy here. Right? You see the whole deal? Now that's an a one two,
a two three, a three one, and does that go with
a plus or a minus? Yeah, now that takes
a minute of thinking, doesn't it, because one
row exchange doesn't get it in line. So what is the answer for this? Plus or minus? Plus, because it
takes two exchanges. I could exchange rows one and
three and then two and three. Two exchanges makes
this thing a plus. And then finally we have --
we're going to have two more. OK. Zero zero a one three, a two
one zero zero, zero a three two zero. And one more guy. Zero zero a one
three, zero a two two zero, A three one zero zero. And let's put down
what we get from those. An a one three, an a two one,
and an a three two, and I think that one is a plus. And this guys is a minus because
one exchange would put it -- would order it. And that's a minus. All right, that has
taken one whole board just to do the three by three. But do you agree
that we now have a formula for the
determinant which came from the three properties? And it must be it. And I'm going to
keep that formula. That's a famous -- that three
by three formula is one that if, if the cameras will follow
me back to the beginning here, I, I get the ones with the plus
sign are the ones that go down like down this way. And the ones with
the minus signs are sort of the ones
that go this way. I won't make that precise. For two reasons, one,
it would clutter up the board, and second reason,
it wouldn't be right for four by fours. For four by four, let
me just say right away, four by four matrix --
the, the cross diagonal, the wrong diagonal happens
to come out with a plus sign. Why is that? If I have a four by
four matrix with ones coming on the counter diagonal,
that determinant is plus. Why? Why plus for that guy? Because if I exchange
rows one and four and then I exchange rows two and
three, I've got the identity, and I did two exchanges. So this down to this, like,
you know, down toward Miami and down toward LA stuff is,
like, three by three only. OK. But I do want to get now -- I don't want to go through
this for a four by four. I do want to get now
the general formula. So this is what I refer to in
the book as the big formula. So now this is the big
formula for the determinant. I'm asking you to make a jump
from two by two and three by three to n by n. OK, so this will
be the big formula. That the determinant of A is
the sum of a whole lot of terms. And what are those terms? And, and is it a
plus or a minus sign, and I have to tell you
which, which it is, because this came in -- in
the three by three case, I had how many terms? Six. And half were plus
and half were minus. How many terms are you
figuring for four by four? If I get two terms in the
two by two case, three -- six terms in the three by three
case, what's that pattern? How many terms in the
four by four case? Twenty-four. Four factorial. Why four factorial? This will be a sum
of n factorial terms. Twenty-four, a
hundred and twenty, seven hundred and twenty,
whatever's after that. OK. Half plus and half minus. And where do those n
factorial -- terms come from? This is the moment to
listen to this lecture. Where do those n
factorial terms come from? They come because the first,
the guy in the first row can be chosen n ways. And after he's chosen, that's
used up that, that column. So the one in the second row
can be chosen n minus one ways. And after she's chosen,
that second column has been used. And then the one in the third
row can be chosen n minus two ways, and after it's chosen -- notice how I'm getting
these personal pronouns. But I've run out. And I'm not willing to
stop with three by three, so I'm just going to
write the formula down. So the one in the first row
comes from some column alpha. I don't know what alpha is. And the one in the -- I multiply that by somebody
in the second row that comes from some different column. And I multiply that by somebody
in the third row who comes from some yet different column. And then in the n-th
row, I don't know what -- I don't know how to draw. Maybe omega, for last. And the whole point
is then that -- that those column
numbers are different, that alpha, beta, gamma, omega,
that set of column numbers is some permutation,
permutation of one to n. It, it, the n column
numbers are each used once. And that gives us
n factorial terms. And when I choose
a term, that means I'm choosing somebody
from every row and column. And then I just -- like the way
I had this from row and column one, row and column two, row
and column three, so that -- what was the alpha beta stuff
in that, for that term here? Alpha was one, beta was
two, gamma was three. The permutation was, was
the trivial permutation, one two three, everybody
in the right order. You see that formula? It's -- do you see why I
didn't want to start with that the first day, Friday? I'd rather we understood
the properties. Because out of this
formula, presumably I could figure out all
these properties. How would I know that the
determinant of the identity matrix was one, for example,
out of this formula? Why is -- if A is
the identity matrix, how does this formula
give me a plus one? You see it, right? Because, because almost
all the terms are zeros. Which term isn't zero, if,
if A is the identity matrix? Almost all the terms are zero
because almost all the As are zero. It's only, the only
time I'll get something is if it's a one one times a
two two times a three three. Only, only the,
only the permutation that's in the right order
will, will give me something. It'll come with a plus sign. And the determinant of
the identity is one. So, so we could go back
from this formula and prove everything. We could even try to prove
that the determinant of A B was the determinant of A
times the determinant of B. But like next week we would
still be working on it, because it's not -- clear from -- if I took A B, my God. You know --. The entries of A B would
be all these pieces. Well, probably, it's probably
-- historically it's been done, but it won't be repeated
in eighteen oh six. OK. It would be possible probably
to see, why the determinant of A equals the determinant
of A transpose. That was another, like,
miracle property at the end. That would, that
would, that's an easier one, which we could find. OK. Is that all right
for the big formula? I could take you
then a, a typical -- let me do an example. Which I'll just create. I'll take a four by four matrix. I'll put some, I'll put some
ones in and some zeros in. OK. Let me -- I don't know how many to
put in, to tell the truth. I've never done this before. I don't know the
determinant of that matrix. So like mathematics is being
done for the first time in, in front of your eyes. What's the determinant? Well, a lot of -- there
are twenty-four terms, because it's four by four. Many of them will be
zero, because I've got all those zeros there. Maybe the whole
determinant is zero. I mean, I -- is that
a singular matrix? That possibility
definitely exists. I could, I could, So one way
to do it would be elimination. Actually, that would probably
be a fairly reasonable way. I could use elimination,
so I could use -- go back to those properties,
that -- and use elimination, get down, eliminate it down,
do I have a row of zeros at the end of elimination? The answer is zero. I was thinking, shall
I try this big formula? OK. Let's try the big formula. How -- tell me one way I can go
down the matrix, taking a one, taking a one from every row and
column, and make it to the end? So it's -- I get
something that isn't zero. Well, one way to do it, I could
take that times that times that times that times that. That would be one and,
and, and I just said, that comes in with what sign? Plus. That comes with a plus sign. Because, because
that permutation -- I've just written
the permutation about four three two
one, and one exchange and a second exchange,
two exchanges puts it in the correct order. Keep walking away, don't.... OK, we're executing a
determinant formula here. Uh as long as it's not
periodic, of course. If he comes back I'm in -- no. All right, all right. OK, so that would
give me a plus one. All right. Are there any others? Well, of course we
see another one here. This times this times this
times this strikes us right away. So that's the order
three, the order -- let me make a little
different mark here. Three two one four. And is that a plus or a
minus, three two one four? Is that, is that permutation
a plus or a minus permutation? It's a minus. How do you see that? What exchange shall I do to
get it in the right order? If I exchange the one and the
three I'm in the right orders, took one exchange to do it,
so that would be a plus -- that would be a minus one. And now I don't know if
there're any more here. Let's see. Let me try again
starting with this. Now I've got to pick somebody
from -- oh yeah, see, you see what's happening. If I I start there, OK,
column three is used. So then when I go to next
row, I can't use that, I must use that. Now columns two
and three are used. When I come to this
row I must use that. And then I must use that. So if I start there, this
is the only one I get. And similarly, if I start there,
that's the only one I get. So what's the determinant? What's the determinant? Zero. The determinant is
zero for that case. Because we, we were able to
check the twenty-four terms. Twenty-two of them were zero. One of them was plus one. One of them was minus one. Add up the twenty-four
terms, zero is the answer. OK. Well, I didn't know
it would be zero, I -- because I wasn't,
like, thinking ahead. I was a little scared, actually. I said, that,
apparition went by. So and I don't know if
the camera caught that. So whether the rest
of the world will realize that I was in danger
or not, we don't know. But anyway, I guess
he just wanted to be sure that we got
the right answer, which is determinant zero. And then that makes me
think, OK, the matrix must be, the matrix
must be singular. And then if the
matrix is singular, maybe there's another way to see
that it's singular, like find something in its null space. Or find a combination of
the rows that gives zero. And like what d- what, what
combination of those rows does give zero. Suppose I add rows
one and rows three. If I add rows one and
rows three, what do I get? I get a row of all ones. Then if I add rows two and rows
four I get a row of all ones. So row one minus row two
plus row three minus row four is probably the zero row. It's a singular matrix. And I could find something in
its null space the same way. That would be a combination
of columns that gives zero. OK, there's an example. All right. So that's, well, that
shows two things. That shows how we get
the twenty-four terms and it shows the great
advantage of having a lot of zeros in there. OK. So we'll use this big
formula, but I want to pick -- I want to go onward
now to cofactors. Onward to cofactors. Cofactors is a way of breaking
up this big formula that connects this n by n -- this
is an n by n determinant that we've just have a formula
for, the big formula. So cofactors is a way to connect
this n by n determinant to, determinants one smaller. One smaller. And the way we want to do it
is actually going to show up in this. Since the three by three is the
one that we wrote out in full, let's, let me do
this three by -- so I'm talking about cofactors,
and I'm going to start again with three by three. And I'm going to take
the, the exact formula, and I'm just going to
write it as a one one -- this is the determinant
I'm writing. I'm just going to say
a one one times what? A one one times what? And it's a one one times
a two two a three three minus a two three a three two. Then I've got the a one
two stuff times something. And I've got the a one
three stuff times something. Do you see what I'm doing? I'm taking our big formula
and I'm saying, OK, choose column -- out of the first row,
choose column one. And take all the possibilities. And those extra
factors will be what we'll call the cofactor, co
meaning going with a one one. So this in parenthesis
are, these are in, the cofactors are in parens. A one one times something. And I figured out what that
something was by just looking back -- if I can walk back
here to the, to the a one one, the one that comes down the
diagonal minus the one that comes that way. That's, those are the two,
only two that used a one one. So there they are, one
with a plus and one with a minus. And now I can write in the -- I could look back and
see what used a one two and I can see what
used a one three, and those will give
me the cofactors of a one two and a one three. Before I do that, what's this
number, what is this cofactor? What is it there that's
multiplying a one one? Tell me what a two two a three
three minus a two three a three two is, for this -- do you recognize that? Do you recognize -- let's see, I can --
and I'll put it here. There's the a one one. That's used column one. Then there's -- the other
factors involved these other columns. This row is used. This column is used. So this the only things
left to use are these. And this formula
uses them, and what's the, what's the cofactor? Tell me what it is because
you see it, and then -- I'll be happy you see what
the idea of cofactors. It's the determinant
of this smaller guy. A one one multiplies
the determinant of this smaller guy. That gives me all the a one
one part of the big formula. You see that? This, the determinant
of this smaller guy is a two two a three three
minus a two three a three two. In other words, once I've
used column one and row one, what's left is all the
ways to use the other n-1 columns and n-1
rows, one of each. All the other -- and that's the
determinant of the smaller guy of size n-1. So that's the whole
idea of cofactors. And we just have to remember
that with determinants we've got pluses and minus
signs to keep straight. Can we keep this
next one straight? Let's do the next one. OK, the next one will
be when I use a one two. I'll have left -- so I can't
use that column any more, but I can use a two one and a
two three and I can use a three one and a three three. So this one gave me a one
times that determinant. This will give me a one two
times this determinant, a two one a three three minus
a two three a three one. So that's all the stuff
involving a one two. But have I got the sign right? Is the determinant of that
correctly given by that or is there a minus sign? There is a minus sign. I can follow one of these. If I do that times
that times that, that was one that's
showing up here, but it should have showed --
it should have been a minus. So I'm going to build that
minus sign into the cofactor. So, so the cofactor
-- so I'll put, put that minus sign in here. So because the cofactor
is going to be strictly the thing that multiplies
the, the factor. The factor is a one two,
the cofactor is this, is the parens, the
stuff in parentheses. So it's got the
minus sign built in. And if I did -- if I went on
to the third guy, there w- there'll be this and
this, this and this. And it would take
its determinant. It would come out
plus the determinant. So now I'm ready to
say what cofactors are. So this would be a plus and a
one three times its cofactor. And over here we had plus a
one one times this determinant. But and there we had the a
one two times its cofactor, but the -- so the point is
the cofactor is either plus or minus the determinant. So let me write that
underneath them. What is the, what are cofactors? The cofactor if any
number aij, let's say. This is, this is all the terms
in the, in the big formula that involve aij. We're especially interested
in a1j, the first row, that's what I've been talking about,
but any row would be all right. All right, so -- what terms involve aij? So -- it's the determinant
of the n minus one matrix -- with row i, column j erased. So it's the, it's a
matrix of size n-1 with -- of course, because I can't use
this row or this column again. So I have the matrix all there. But now it's multiplied
by a plus or a minus. This is the cofactor, and
I'm going to call that cij. Capital, I use
capital c just to, just to emphasize that
these are important and emphasize that
they're, they're, they're different from the (a)s. OK. So now is it a plus
or is it a minus? Because we see
that in this case, for a one one it was a plus,
for a one two I -- this is ij -- it was a minus. For this ij it was a plus. So any any guess on the
rule for plus or minus when we see those examples,
ij equal one one or one three was a plus? It sounds very like
i+j odd or even. That, that's
doesn't surprise us, and that's the right answer. So it's a plus if i+j is even
and it's a minus if i+j is odd. So if I go along row one
and look at the cofactors, I just take those determinants,
those one smaller determinants, and they come in order plus
minus plus minus plus minus. But if I go along row two and,
and, and take the cofactors of sub-determinants, they
would start with a minus, because the two one entry,
two plus one is odd, so the -- like there's a pattern plus
minus plus minus plus if it was five by five, but then if I was
doing a cofactor then this sign would be minus plus minus
plus minus, plus minus plus -- it's sort of checkerboard. OK. OK. Those are the signs that,
that are given by this rule, i+j even or odd. And those are built
into the cofactors. The thing is called
a minor without th- before you've built in the sign,
but I don't care about those. Build in that sign and
call it a cofactor. So what's the cofactor formula? OK. What's the cofactor
formula then? Let me come back to
this board and say, what's the cofactor formula? Determinant of A is -- let's go along the first row. It's a one one
times its cofactor, and then the second guy is a
one two times its cofactor, and you just keep going
to the end of the row, a1n times its cofactor. So that's cofactor for -- along row one. And if I went along row I, I
would -- those ones would be Is. That's worth putting a box over. That's the cofactor formula. Do you see that -- actually, this would
give me another way I could have started the
whole topic of determinants. And some, some people
might do it this -- choose to do it this way. Because the cofactor
formula would allow me to build up an
n by n determinant out of n-1 sized determinants, build
those out of n-2, and so on. I could boil all the
way down to one by ones. So what's the cofactor formula
for two by two matrices? Yeah, tell me that. What's the cofactor for us? Here is the, here is the world's
smallest example, practically, of a cofactor formula. OK. Let's go along row one. I take this first guy
times its cofactor. What's the cofactor
of the one one entry? d, because you strike out
the one one row and column and you're left with d. Then I take this guy,
b, times its cofactor. What's the cofactor of b? Is it c or it's -- minus c, because I
strike out this guy, I take that determinant, and
then I follow the i+j rule and I get a minus, I get an odd. So it's b times minus c. OK, it worked. Of course it, it worked. And the three by three works. So that's the cofactor formula,
and that is, that's an -- that's a good formula to know,
and now I'm feeling like, wow, I'm giving you a lot of
algebra to swallow here. Last lecture gave
you ten properties. Now I'm giving you -- and by the way, those ten
properties led us to a formula for the determinant
which was very important, and I haven't
repeated it till now. What was that? The, the determinant is
the product of the pivots. So the pivot formula
is, is very important. The pivots have all this
complicated mess already built in. As you did elimination
to get the pivots, you built in all this horrible
stuff, quite efficiently. Then the big formula with
the n factorial terms, that's got all the
horrible stuff spread out. And the cofactor formula
is like in between. It's got easy stuff times
horrible stuff, basically. But it's, it shows you,
how to get determinants from smaller determinants, and
that's the application that I now want to make. So may I do one more example? So I remember the general idea. But I'm going to use this
cofactor formula for a matrix -- so here is going
to be my example. It's -- I promised in
the, in the lecture, outline at the very
beginning to do an example. And let me do -- I'm going to pick
tri-diagonal matrix of ones. I could, I'm drawing
here the four by four. So this will be the matrix. I could call that A4. But my real idea
is to do n by n. To do them all. So A -- I could -- everybody understands
what A1 and A2 are. Yeah. Maybe we should just do A1
and A2 and A3 just for -- so this is What's the determinant of A1? A4. What's the determinant of A1? So, so what's the matrix
A1 in this formula? It's just got that. So the determinant is one. What's the determinant of A2? So it's just got this two by
two, and its determinant is -- zero. And then the three by three. Can we see its determinant? Can you take the determinant
of that three by three? Well, that's not quite so
obvious, at least not to me. Being three by three,
I don't know -- so here's a, here's
a good example. How would you do that
three by three determinant? We've got, like, n
factorial different ways. Well, three factorial. So we've got six ways. OK. I mean, one way to do it -- actually the way
I would probably do it, being three
by three, I would use the complete the big formula. I would say, I've
got a one from that, I've got a zero from that, I've
got a zero from that, a zero from that, and this
direction is a minus one, that direction's a minus one. I believe the
answer is minus one. Would you do it another way? Here's another way
to do it, look. Subtract row three from -- I'm just looking
at this three by three. Everybody's looking
at the three by three. Subtract row three from row two. Determinant doesn't change. So those become zeros. OK, now use the
cofactor formula. How's that? How can, how -- if this was now
zeros and I'm looking at this three by three, use
the cofactor formula. Why not use the cofactor
formula along that row? Because then I take that
number times its cofactor, so I take this number -- let
me put a box around it -- times its cofactor, which is the
determinant of that and that, which is what? That two by two matrix
has determinant one. So what's the cofactor? What's the cofactor
of this guy here? Looking just at
this three by three. The cofactor of that
one is this determinant, which is one times negative. So that's why the answer
came out minus one. OK. So I did the three by three. I don't know if we want
to try the four by four. Yeah, let's -- I guess that
was the point of my example, of course, so I have to try it. Sorry, I'm in a good
mood today, so you have to stand for all the bad jokes. OK. OK. So what was the matrix? Ah. OK, now I'm ready
for four by four. Who wants to -- who wants
to guess the, the -- I don't know, frankly,
this four by four, what's, what's the determinant. I plan to use cofactors. OK, let's use cofactors. The determinant of A4 is -- OK, let's use cofactors
on the first row. Those are easy. So I multiply this number,
which is a convenient one, times this determinant. So it's, it's one times
the, this three by three determinant. Now what is -- do you
recognize that matrix? It's A3. So it's one times the
determinant of A3. Coming along this row is a
one times this determinant, and it goes with a plus, right? And then we have this one. And what is its cofactor? Now I'm looking at, now
I'm looking at this three by three, this three
by three, so I'm looking at the three by three
that I haven't X-ed out. What is that -- oh, now
it, we did a plus or a -- is it plus this determinant,
this three by three determinant, or minus it? It's minus it, right,
because this is -- I'm starting in a one two
position, and that's a minus. So I want minus
this determinant. But these guys are X-ed out. OK. So I've got a three by three. Well, let's use cofactors again. Use cofactors of the
column, because actually we could use cofactors
of columns just as well as rows, because,
because the transpose rule. So I'll take this one, which
is now sitting in the plus position, times its
determinant -- oh! Oh, hell. What -- oh yeah, I
shouldn't have said hell, because it's all right. OK. One times the determinant. What is that matrix
now that I'm taking the, this smaller one of? Oh, but there's a minus, right? The minus came
from, from the fact that this was in the one
two position and that's odd. So this is a minus one
times -- and what's -- and then this one
is the upper left, that's the one one position
in its matrix, so plus. And what's this matrix here? Do you recognize that? That matrix is -- yes, please say it -- A2. And we -- that's our
formula for any case. A of any size n is equal to the
determinant of A n minus one, that's what came from taking
the one in the upper corner, the first cofactor, minus the
determinant of A n minus two. What we discovered
there is true for all n. I didn't even mention
it, but I stopped taking cofactors when I got this one. Why did I stop? Why didn't I take the
cofactor of this guy? Because he's going to get
multiplied by zero, and no, no use wasting time. Or this one too. The cofactor, her
cofactor will be whatever that
determinant is, but it'll be multiplied by zero,
so I won't bother. OK, there is the formula. And that now tells us -- I could figure out
what A4 is now. Oh yeah, finally I can get A4. Because it's A3, which is minus
one, minus A2, which is zero, so it's minus one. You see how we're
getting kind of numbers that you might not have guessed. So our sequence right now is
one zero minus one minus one. What's the next one
in the sequence, A5? A5 is this minus
this, so it is zero. What's A6? A6 is this minus
this, which is one. What's A7? I'm, I'm going to be stopped
by either the time runs out or the board runs out. OK, A7 is this minus
this, which is one. I'll stop here, because time
is out, but let me tell you what we've got. What -- these determinants
have this series, one zero minus one
minus one zero one, and then it starts repeating. It's pretty fantastic. These determinants
have period six. So the determinant
of A sixty-one would be the determinant
of A1, which would be one. OK. I hope you liked that example. A non-trivial example of a
tri-diagonal determinant. Thanks. See you on Wednesday.
At 24:30
Maybe the Executioner wanted to axe a question.
You can see the prof's shit-eating grin when he says "executing a determinant formula" right after.
oh god.. flashbacks..
the executioner didn't help either.
Foreshadowing...
The prof didn't miss a beat!