<s>[INST] Problem when row reducing matrix with complex entries I have to find a row-reduced matrix which is row-equivalent to
$$\left[\begin{array}{ccc} i & -(1+i) & 0 \\ 1 & -2 & 1 \\ 1 & 2i & -1\\ \end{array}\right]$$
On one hand
$$\left[\begin{array}{ccc} i & -(1+i) & 0 \\ 1 & -2 & 1 \\ 1 & 2i & -1\\ \end{array}\right] \\ \xrightarrow{R_1 \leftrightarrow R_2} \left[\begin{array}{ccc} 1 & -2 & 1 \\ i & -(1+i) & 0 \\ 1 & 2i & -1\\ \end{array}\right]$$
$$\xrightarrow{R_2 \gets -iR_1 + R_2} \left[\begin{array}{ccc} 1 & -2 & 1 \\ 0 & -1+i & -i \\ 1 & 2i & -1\\ \end{array}\right] $$
$$\xrightarrow{R_3 \gets -R_1 + R_3} \left[\begin{array}{ccc} 1 & -2 & 1 \\ 0 & -1+i & -i \\ 0 & 2+2i & -2\\ \end{array}\right] $$
$$\xrightarrow{\ \frac{1}{(-1+i)}R_2} \left[\begin{array}{ccc} 1 & -2 & 1 \\ 0 & 1 & 1/(-1-i) \\ 0 & 2+2i & -2\\ \end{array}\right] $$
$$\xrightarrow{\ R_1\gets 2R_2+R_1} \left[\begin{array}{ccc} 1 & 0 & i \\ 0 & 1 & 1/(-1-i) \\ 0 & 2+2i & -2\\ \end{array}\right] $$
$$\xrightarrow{\ R_3\gets -(2+2i)R_2+R_3} \left[\begin{array}{ccc} 1 & 0 & i \\ 0 & 1 & 1/(-1-i) \\ 0 & 0 & -4\\ \end{array}\right] $$
and this last matrix is equivalent to the identity matrix $I_{3x3}$.
On the other hand I reduced it with other row operations and concluded the same as WOLFRAM ALPHA. Which one is wrong? Any help is appreciated.
 [/INST] Your mistake is in 4th step. When you are taking $(-1+i)$ common from $R_2$. Then element at 2nd row and 3rd column is wrong I think.
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