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- Let omega be the complex number,
- Cosine of 2 pie over 3 + i Sin of 2 pie over 3
- Then the number of distinct complex numbers,
- z satisfying with this determinant equaling zero
- so lets have the 3 by 3 determinant equaling zero,
- so lets just evaluate this determinant and see if we
- can solve for z or figure out how many
- complex number z we get satisfying this with
- esentially an equation, so lets evaluate the determinant
- so if we start with this term up here
- it's going to be this term times the determinant
- of the sub 2 by 2 matrix, so it's going to be z + 1,
- z+1 times z + omega squared, times z + omega,
- minus 1, so this is the determinant of the sub,
- of the sub 2 by 2 matrix right over here, the sub
- 2 by 2 derterminant and this is z + 1 and then we're
- going to, we have to swap signs, we go here,
- we want to put a negative sign out front,
- we have a kind of checker-board pattern when you
- evaluate the determinants, so minus omega times,
- you black out that row, that column, so omega
- the sub determinant is omega times z + omega,
- let me just multiply that out right now, so that's
- z omega + omega squared, minus omega squared times 1,
- so minues omega squared, so thats pretty,
- 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,
- omgea times 1, omega times 1 is omega,
- minus omega squared times z + omega,
- so minues omega squared z, omega squared z,
- minus omega to the 4th.
- And remember this whole expression needs to be equal to zero,
- this whole thing needs to be equal to zero,
- now lets see if we can, lets 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.
- Then we have this minus 1,
- then all of that is going to be
- multiplied by z + 1, let me just continue with these
- blue parts since we're already focused here.
- So this is going to be z times all of this,
- so it's going 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, minus 1, and then let's simplify this over here,
- so in the green, these canceled out,
- so we're just left with, I'll do this in the green,
- negative omega times z omega, so it's minus z omega squared,
- omega, and then over here in magenta,
- here in magenta we have plus, plus omega to the third power,
- omega to the third power, minus omega to the third z,
- right, oh let me be careful, that would be omega,
- omega to the 4th, omega square times omega squared is
- omega to the 4th, omega to the4th z, and then
- omega squared times omega to the 4th is omega to the 6th,
- and we have a negative sign, so negative omega to the 6th,
- and of course this whole thing needs to be equal to zero.
- Now there's probably, let's see if we can simpliy this,
- so let's just group the terms, the different powers of z,
- so we have a z to the 3rd, a z to the 3rd,
- z to the 3rd, and that is the only z to the 3rd we have,
- and then let's group, let's group the z squared terms,
- 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 co-efficients,
- this is going to be the same thing as omega squared,
- that's this one right over here, plus omega,
- plus omega, plus 1, plus 1, times, 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,
- the z terms, I'll do it in this color, so this is a z term,
- we're just multiplting 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 3 over here,
- so we have plus omega to the 3rd, and then we have
- plus omega to the third, oh wait, this color is too close,
- plus omega to the 3rd, and our next z is minus 1,
- but we actually have this omega over here,
- so plus omega - 1, I was getting confused,
- because this white is so close to this pink,
- so it's omega to the 3rd, omega to the 3rd,
- minus 1 + omega, times z, and then.
- Oh! We can't forget this, we have this term
- over here, we have omega to the 4th times z,
- omega to the 4th times z, so we have minus 1,
- right, omega to the 3rd z, then we have minus 1 z
- then we have, then we have plus omega z, and then
- we have negative omega to the 4th z,
- so I'll just put it out here, I'm not doing in the
- nice descending power order, but I don't want to
- have to re-write all of this.
- So we took care of all the z terms, and then what
- we have left is what we can kind of call the
- cost of terms from the z point of view,