<br />
\left(<br />
\begin{array}{xyzk}<br />
3 & i & 2+i & 3i\\<br />
-i & 1 & 1 & 1\\<br />
1 & 1 & 2+i & 1<br />
\end{array}<br />
\right)<br /><br />
\ = \left(<br />
\begin{array}{xyzk}<br />
3 & i & 2+i & 3i\\<br />
1 & i & i & i\\<br />
1 & 1 & 2+i & 1<br />
\end{array}<br />
\right)<br />
Multiply Row 2 by i, remember you can multiply by any constant - and i is just \sqrt{-1}. i*i = -1, i*i*i = -i, i*i*i*i = 1
<br />
\ = \left(<br />
\begin{array}{xyzk}<br />
2 & 0 & 2 & 2i\\<br />
1 & i & i & i\\<br />
1 & 1 & 2+i & 1<br />
\end{array}<br />
\right)<br /><br />
\ = \left(<br />
\begin{array}{xyzk}<br />
2 & 0 & 2 & 2i\\<br />
0 & i-1 & -2 & i-1\\<br />
1 & 1 & 2+i & 1<br />
\end{array}<br />
\right)<br /><br />
\ = \left(<br />
\begin{array}{xyzk}<br />
1 & 0 & 1 & i\\<br />
0 & i-1 & -2 & i-1\\<br />
1 & 1 & 2+i & 1<br />
\end{array}<br />
\right)<br />
R1 = R1 - R2 and R2 = R2 - R3 and R1 = 1/2 R1
<br />
\ = \left(<br />
\begin{array}{xyzk}<br />
1 & 0 & 1 & i\\<br />
-1 & i-1 & -3 & -1\\<br />
1 & 1 & 2+i & 1<br />
\end{array}<br />
\right)<br /><br />
\ = \left(<br />
\begin{array}{xyzk}<br />
1 & 0 & 1 & i\\<br />
0 & i & i-1 & 0\\<br />
1 & 1 & 2+i & 1<br />
\end{array}<br />
\right)<br /><br />
\ = \left(<br />
\begin{array}{xyzk}<br />
1 & 0 & 1 & i\\<br />
0 & -1 & -1-i & 0\\<br />
1 & 1 & 2+i & 1<br />
\end{array}<br />
\right)<br />
R2 = R2 - R1 and R2 = R2 + R3 and R2 = i*R2
<br />
\ = \left(<br />
\begin{array}{xyzk}<br />
1 & 0 & 1 & i\\<br />
0 & -1 & -1-i & 0\\<br />
0 & 1 & 1+i & 1-i<br />
\end{array}<br />
\right)<br /><br />
\ = \left(<br />
\begin{array}{xyzk}<br />
1 & 0 & 1 & i\\<br />
0 & 0 & 0 & 1-i\\<br />
0 & 1 & 1+i & 1-i<br />
\end{array}<br />
\right)<br />
R3 = R3 - R1 and R2 = R2 + R3
<br />
\ = \left(<br />
\begin{array}{xyzk}<br />
1 & 0 & 1 & i\\<br />
0 & 0 & 0 & 1-i\\<br />
0 & 1 & 1+i & 1-i<br />
\end{array}<br />
\right)<br />
You swap Row2 and Row3 and you get:
<br />
\ = \left(<br />
\begin{array}{xyzk}<br />
1 & 0 & 1 & i\\<br />
0 & 1 & 1+i & 1-i\\<br />
0 & 0 & 0 & 1-i<br />
\end{array}<br />
\right)<br /><br />
\ = \left(<br />
\begin{array}{xyzk}<br />
1 & 0 & 1 & i\\<br />
0 & 1 & 1+i & 0\\<br />
0 & 0 & 0 & 0<br />
\end{array}<br />
\right)<br />
R2 = R2 - R3 and you can eliminate Col 4 of R3
And viola! You end up with:
x + z = i
y + (1+i)z = 0