Finding Solution Set for Vectors

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To determine the solution set for the given system of equations involving complex numbers, perform row-reduction to achieve row echelon form. This process is similar to handling real number matrices, with the added flexibility of multiplying rows by complex numbers. An effective method includes manipulating the equations, such as subtracting multiples of one equation from another to eliminate variables. Solving the equations can also be approached using traditional methods learned in linear algebra. Ultimately, applying these techniques will yield the solution set for the vectors involved.
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How can I determine the solution set?

(1+2i)x1 + (1-i)x2 + x3 = 0,
ix1 + (1+i)x2 - ix3 = 0,
2ix1 + ix2 + (1+3i)x3 = 0.

Thanks..
 
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Simply perform row-reduction to reduce the system to its row echelon form. The only difference with this and matrices containing only real numbers is that you can now multiply row by complex numbers as well.
 
Or just solve the three equations the way you learned to solve linear equations way back. For example, subtracting 2 times the second equation from the third eliminates x1.
 

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