Gaussian Elimination, Question Check. I'm going through my practice problems, and just want to know if I am doing this right:
$$ 2x_1 - 3x_2 = -2$$
$$ 2x_1 + x_2 = 1$$
$$ 3x_1 + 2x_2 = 1$$
And this is my solution:
$$
\begin{align}
\begin{bmatrix}
2 & 3 & -2\\
2 & 1 & 1\\
3 & 2 & 1
\end{bmatrix}
\end{align}
$$
$$
\begin{align}
\begin{bmatrix}
1 & -3/2 & -1\\
2 & 1 & 1\\
3 & 2 & 1
\end{bmatrix}
\end{align}
$$
$$
\begin{align}
\begin{bmatrix}
1 & -3/2 & -1\\
0 & -1 & -1\\
3 & 2 & 1
\end{bmatrix}
\end{align}
$$
$$
\begin{align}
\begin{bmatrix}
1 & -3/2 & -1\\
0 & -1 & -1\\
0 & -1 & -2
\end{bmatrix}
\end{align}
$$
$$
\begin{align}
\begin{bmatrix}
1 & -3/2 & -1\\
0 & 1 & 1\\
0 & -1 & -2
\end{bmatrix}
\end{align}
$$
$$
\begin{align}
\begin{bmatrix}
1 & -3/2 & -1\\
0 & 1 & 1\\
0 & 0 & -1
\end{bmatrix}
\end{align}
$$
Can anyone tell me if I did this right, or if I did a mistake where? Thanks.

first  consider $2x_1 - 3x_2 = -2$ and $2x_1 + x_2 = 1$ your solution is $x_1=\frac{1}{8} , x_2=\frac{3}{4}$ 
now consider :
$2x_1 + x_2 = 1$ and $3x_1 + 2x_2 = 1$ your solution is $x_1=1,x_2=-1$ 
so your system does not an answer
and 3 th row of  your last matrix \begin{align}
$\begin{bmatrix}
1 & -3/2 & -1\\
0 & 1 & 1\\
0 & 0 & -1
\end{bmatrix}
\end{align}$
say that your matrix  has not answer too  
($ 0x_1+0x_2=1 $ is impossible)
