System of linear equations with complex coefficients

[Qubix]

1. You have a system of equations of the following form:
x + iy = 0
-ix + z = 0
y - z = 0

or

-2Sqr(5)x - iy = 0
ix - 2Sqr(5)y + 2iz = 0
-2iy - 2Sqr(5)z = 0



2. What is the general way in which I can solve such a system? I've tried Kramer's Rule, but it does not seem to work since Dx, Dy, Dz all give zero.

 

[rock.freak667]

1. You have a system of equations of the following form:
x + iy = 0
-ix + z = 0
y - z = 0

Quite easily it can be seen that (0,0,0) is a solution here

What you can do is form an augmented matrix and row reduce.

or take the first equation x+iy=0 and put it such that x= -iy and put that into the other equations that have 'x' in it. You will now have two equations with two unknowns.

Do the same with the other set of equations.

 

[HallsofIvy]

. From -ix+ z= 0, ix= z so x= -iz. We also have x= -iy so -iy= -iz or y= z. The last equation is y- z= 0 which says y= z also. That's why all the determinant are 0: this system has an infinite number of solutions. You can choose z to be anything you want and write x and y in terms of z.

 

[Qubix]

Thank you very much for your answers