row exchange results in wrong answer? This is probably a very simple, but why is it that when I do row exchanges I end up with the wrong answer.  I have the matrix equation $Ax = B$, as:
$$
\underbrace{
\begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & -2 \\ -1 & -1 & 1 \end{bmatrix}
}_A
\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}
=
\underbrace{
\begin{bmatrix} 3 \\ 1 \\ -6 \end{bmatrix}
}_B$$
I know the answer is $x = \begin{bmatrix} 7 \\ -3 \\ -2 \end{bmatrix}$, but when I do a row exchange (whether needed or not) I end up with the wrong answer.  Why?  NOTE: I exchanged rows 2 and 3.

$$\left[ \begin{array} {rrr|r}
 1 &  0 &  2 &  3 \\
 0 &  1 & -2 &  1 \\
-1 & -1 &  1 & -6 \\
\end{array}\right]$$
Exchange row 2 and 3 :
$$\left[ \begin{array} {rrr|r}
 1 &  0 &  2 &  3 \\
-1 & -1 &  1 & -6 \\
 0 &  1 & -2 &  1 \\
\end{array}\right]$$
Add row 1 to row 2:
$$\left[ \begin{array} {rrr|r}
 1 &  0 &  2 &  3 \\
 0 & -1 &  3 & -3 \\
 0 &  1 & -2 &  1 \\
\end{array}\right]$$
Add row 2 to row 3:
$$\left[ \begin{array} {rrr|r}
 1 &  0 &  2 &  3 \\
 0 & -1 &  3 & -3 \\
 0 &  0 &  1 & -2 \\
\end{array}\right]$$
Multiply row 2 by -1:
$$\left[ \begin{array} {rrr|r}
 1 &  0 &  2 &  3 \\
 0 &  1 & -3 &  3 \\
 0 &  0 &  1 & -2 \\
\end{array}\right]$$
Add 3 times row 3 to row 2:
$$\left[ \begin{array} {rrr|r}
 1 &  0 &  2 &  3 \\
 0 &  1 &  0 & -3 \\
 0 &  0 &  1 & -2 \\
\end{array}\right]$$
Add -2 times row 3 to row 1:
$$\left[ \begin{array} {rrr|r}
 1 &  0 &  0 &  7 \\
 0 &  1 &  0 & -3 \\
 0 &  0 &  1 & -2 \\
\end{array}\right]$$
