Test the following systems as in Exercise 2. \[ \begin{array}{rl} 2x_1 + (-1+i)x_2 + x_4 = 0, & \left(1+\frac{i}{2}\right)x_1 + 8x_2 - ix_3 - x_4 = 0, \\ 3x_2 - 2ix_3 + 5x_4 = 0, & \frac{2}{3}x_1 - \frac{1}{2}x_2 + x_3 + 7x_4 = 0. \end{array} \]
Solution
The two systems are not equivalent.
1. The vector\[ (x_1,x_2,x_3,x_4)=(1+i,2i,3,0) \]satisfies both equations of the first system.
For the first equation, \[ 2(1+i)+(-1+i)(2i)+0=2+2i-2i-2=0. \] For the second, \[ 3(2i)-2i(3)+5(0)=0. \]
2. The same vector does not satisfy the second system.
Substitution into its second equation gives \[ \frac23(1+i)-\frac12(2i)+3+7(0) =\frac{11}{3}-\frac{i}{3}\ne0. \]
3. Hence the systems are not equivalent.
Equivalent systems have exactly the same solution set, contradicting 1 and 2.