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Let omega be the complex
number cosine of 2 pi over 3 plus i sine of 2 pi
over 3, then the number of distinct complex numbers
z satisfying this determinant equaling 0. So we have this 3 by 3
determinant equaling 0. So let's just evaluate
this determinant and see if we can solve
for z, or figure out how many complex numbers z
that we get satisfying this-- what's essentially an equation. So let's evaluate
the determinant. So if we start with
this term up here it's going to be this term times
the determinant of this sub 2 by two matrix. So it's going to be
z plus 1, times z plus omega squared, times
z plus omega, minus 1. So this is the determinant
of the sub 2 by 2 matrix right over here-- the sub 2
by 2 determinant-- and that's the z plus 1. And then we have to swap signs. We go here. We want to put a
negative sign out front. You have that
checkerboard pattern when you evaluate determinants. So minus omega times-- you
block out that row, that column. So the sub-determinant is
omega times z plus omega. Let me just multiply
that out right now. So that's z omega plus omega
squared, minus omega squared, times 1. So that's pretty neat. We already got something
that simplifies a little bit. And then we have to worry
about this omega squared. So it's going to be plus omega
squared times omega times one-- omega times 1
is omega-- minus omega squared times z plus omega. So minus omega squared z
minus omega to the fourth. And remember this
whole expression needs to be equal to 0. This whole thing needs
to be equal to 0. Now let's see if we can simplify
this thing a little bit. So let me first multiply
these two guys in here. So z times z is z squared. Z times omega is z omega. Omega squared times
z is omega squared z. And then omega
squared times omega is omega to the third power--
plus omega to the third power. And then we have
this minus 1 here. And then all of that's going
to be multiplied by z plus 1. Let me just continue
with this blue part since we're already
focused here. So this is going to be
z times all of this. So it's going to be z to
the third power plus z squared omega, plus
omega squared z squared, plus omega to the
third z minus z. I just multiplied z
times all of this. And then plus 1
times all of this. So plus this thing again,
z squared plus z omega, plus omega squared z, plus
omega to the third, minus 1. And then let's simplify
this over here. So that in green,
these canceled out. So we're just left
with-- I'll do the same green-- negative
omega times z omega. So it's minus z omega squared. And then over here
in magenta we have plus omega to the third power
minus omega to the third z. Oh, let me be careful. That would be omega
to the fourth. Omega squared
times omega squared is omega to the fourth--
omega to the fourth z. And then omega squared
times omega to the fourth is omega to the sixth. And we have a negative sign. So negative omega to the sixth. And of course, this whole
thing needs to be equal to 0. Well, let's see if
we can simplify this. Let's just group the terms--
the different powers of z. So we have a z to
the third here. And that is the only z to
the third that we have. And then let's group
the z squared terms. So this is a z squared term. This is a z squared term. That's a z squared term. And then we don't have
any more z squared terms. And so the
coefficients-- this is going to be the same thing as
omega squared-- that's this one right over here-- plus omega,
plus 1, times z squared. So we've taken
care of everything that I've underlined
in pink over here. Now let's worry
about the z terms. I'll do it in this color. So this is a z term. We're just multiplying times z. This is a z term. This is a z term. That is also a z term. And then do we have
any other z terms? Well, this is a z
term right over here. And in fact, these
two cancel out. You have negative
z omega squared, positive z omega squared. So this and this cancel out. And so our only z terms
are these three over here. So we have plus
omega to the third. And then we have plus
omega to the third. Oh let me-- this
color is too close. Plus omega to the third. And then our next z is minus 1. But then we actually
have this omega here. So plus omega minus 1. I was getting confused
because this white is so close to this pink. So it's omega to the third
minus 1, plus omega, times z. And then oh, we
can't forget this. We have this term over here. Omega to the fourth times z. So we have minus 1
omega to the third z. Then we have minus 1 z. Then we have plus omega z. And then we have negative
omega to the fourth z. So I'll just put it out here. I'm not doing it in a nice
descending power order, but I don't want to have
to rewrite all of this. So then we took care
of all of the z terms. And then what we have
left is just what we can view-- the constant
terms, from a z point of view. So we have omega--
So plus omega cubed. Actually, we have
two omega cubed. We have an omega cubed here
and an omega cubed there. We've taken care of those. And then we have minus
omega to the sixth. And then we have minus 1. And all of this
needs to equal 0. Now, this might look
pretty daunting, especially when omega looks
like this fairly complex-- well, it is a complex expression. But maybe, or probably,
it popped into your mind that this can also be expressed
as an exponential using Euler's formula. We know that e to the i theta is
equal to cosine of theta plus i sine of theta. So omega can be
rewritten as, omega is the exact same thing
as e to the 2 pi over 3 i. That's what omega is. And what is e to
the 2 pi over 3 i? We can evaluate that here. It's cosine 2 pi over 3. And we can get our
unit circle out. 2 pi over 3 corresponds
to 120 degrees. So let me get my
unit circle out just so we can figure out
what the values are. And the reason why I'm
doing this-- I'm actually going to evaluate it here
just to get a number here-- but the reason I'm doing this
is it becomes much easier to take the powers of
e to the 2 pi over 3 i than to take the powers
of this thing over here. But if we take the
unit circle, if we're at 120 degrees on the
unit circle like this, this is 2 pi over 3. Then we have a situation. This angle over here is
going to be 60 degrees. And so our height is going to
be square root of 3 over 2. And our x-coordinate is
going to be negative 1/2. So this is going to be
equal to negative 1/2. Cosine of 2 pi over
3 is negative 1/2. And then our sine of 2 pi over
3 is square root of 3 over 2. But that's being
multiplied by an i. So plus square
root of 3 over 2 i. So that is omega. Now, let's think
about omega squared. Omega squared is
going to be this raised to the second power. So it's e to the-- we just
multiply it by 2. e to the 4 pi over 3 i. We just multiply this by 2. So it's equivalent
to 240 degrees. So it's equivalent to
this right over here. So let me write it. This is the same thing
as cosine of 4 pi over 3, plus i sine of four pi over 3. And here our x-coordinate
is going to be the same. It's going to be negative 1/2. So this is equal
to negative 1/2. And our y-coordinate is negative
square root of 3 over 2 i. Now let's think of omega
to the third power. Omega to the third power is
equal to e to the 2 pi i, which is equal to cosine of 2
pi plus i sine of 2 pi. Well, sine of 2 pi is just 0. Cosine of 2 pi is equal to 1. Then we go to omega
to the fourth power, because that also shows
up in these expressions. So we have omega to the
fourth power is equal to-- we just multiply it times 4--
it's e to the 8 pi over 3 i. 8 pi over 3 is equivalent
to-- so this was 2 pi over 3. This is 4 pi over 3. This is essentially 6 pi over 3. And then 8 pi over 3
becomes-- is essentially the same angle as 2 pi
over 3-- so its value is going to be the same. Negative 1/2 plus square
root of 3 over 2 i. And then we don't have
an omega to the fifth, but we do have an omega
to the sixth here. So let's just figure that out. Omega to the sixth-- well,
that's just omega cubed squared. And omega cubed is 1. So it's just going
to be 1 squared, or it's also going to be 1. Now with that out of
the way, let's see what these things simplify to. Hopefully they
simplify to something. So this expression
right here-- omega squared-- we have--
I'll write it up here. We have negative 1/2 minus
square root of 3 over 2 i. That's omega squared. Then we have plus omega. Omega is right here. Negative 1/2 plus square
root of 3 over 2 i. And then we have a plus 1. So the negative square
root of 3 over 2 i and the positive square root
of 3 over 2 i cancel out. Negative 1/2 minus 1/2 is
negative one, plus one. Those cancel out. So this whole expression
comes out to be 0. So we can ignore. This is going to be
0 no matter what. Now let's look at
this over here. This is omega cubed. Omega cubed we know
is 1, plus omega, which is-- so I
should write minus 1/2 plus square root of
3 over 2 i, minus 1, minus omega to the fourth. Well, omega to the fourth
is the same thing as omega. Omega to the fourth is
the same thing as omega. So you have omega
minus something that's the same thing as itself. Those are going to cancel out. Omega cubed is 1. So the 1 and the 1 are
going to cancel out. So this is also equal to 0. I think I see a pattern here. And then finally, let's look
at these characters over here. 2 times omega
cubed-- this is just going to be a 2 here--
minus omega to the sixth. Well, that's just minus 1. And then you have a minus 1. So this 2, minus 1, minus 1. This is also equal to 0. So that whole determinant that
whole equation has simplified to z to the third
power is equal to 0. And the only number,
that when they take it to the third power-- real,
or complex, or anything-- is going to be 0. z equals 0 is the only solution. But they're not asking us that. They're asking the number
of distinct complex numbers z satisfying this. So z equals 0 is the only
one that satisfies this. So the number of complex
numbers satisfying it is 1.