Solve a basic solution and the column manipulation Solve a basic solution.
\begin{align*}A=\begin{array}{cccc} 1 & 1 & -1 & 1 \\ 1 & -1 & -1 & 1 \\ 2 & 1 & 1 & 3 \\\end{array};\end{align*}
here eliminate First element of row2 and row3. 
\begin{align*}\left(\begin{array}{cccc} 1 & 1 & -1 & 1 \\ 0 & -2 & 0 & 0 \\ 0 & -1 & 3 & 1 \\\end{array}\right)\end{align*}
here I can directly calculate for a basic solution. $x_2= 0,x_1=x_3-x_4$ so, the solution vector is $\{4,0,1,-3\}$
However, here seems I've made a wrong calculation(something conflicts with Determinant?)
I add column1 to column 2, column 3, column 4.
\begin{align*}\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & -2 & 0 & 0 \\ 0 & -1 & 3 & 1 \\\end{array}\right)\end{align*}
Here $x_1=0$, then it's wrong answer? Where goes wrong and why, cant do the column manipulation?

Only row operations preserve the solution space, column operations do not.
