複數行列式例題 (英)

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複數行列式例題 (英) : 2010 IIT JEE Paper 1 Problem 53 複數行列式

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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,

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