The Gauss-Jordan elimination method described in section 9.2, works as follows

Suppose you want to solve:
ax+by+cz=0
dx+ey+fz=0
gx+hy+iz=0

where x,y,z are unkowns and a...i are given complex numbers, you first try
to make this system "triangular". First step is to make d and g zero by
taking the first equation and multiplyiing it by -d/a and -g/a
respectively and adding it to the second and third equation respectively.
(I'm assuming a is not zero; if it is just swap the first equation with
either 2nd or 3rd so the coefficient of x is not 0)So you get:
ax+by+cz=0
   Ey+Fz=0
   Hy+Iz=0

for suitable complex numbers E,F,H,I.

My claim is that in our problem we are done! In principle you should now
try to cancel H by taking the 2nd equation and multiplying it by -H/E and
adding it to the thrid. However, we know in advance what will happpen: The
third equation must become 0=0! This is because we are trying to find a
nonzero eigenvector, and we know that there MUST be such a nonzero
solution. So if we would not get 0=0, then you see that the zero solution
would be the only possible solution (namely the last eqaution would read
Kz=0 for some nonzero K, so that implies z=0, and then by
backsubstitution into 2nd equation you get y=0 and into first you get
x=0, a contradiction!)

So where does that leave us? We only need to solve:
ax+by+cz=0
      Ey+Fz=0

Just PICK y=1, solve 2nd equation for z and backsubstitute in first
equation to solve for x. DONE.
[you could have picked z=1, solved for y etc as well]