Whats wrong with my matrix calculations? I am trying to solve this matrix for $(x_1, x_2, x_3)$
$$\pmatrix{
    -1 & -2& -3 \\
     3 & 2 & 1 \\
1& 1 & 1 \\
    2 & 4& 1 
    }* 
\pmatrix{
   x_1\\
x_2 \\
x_3 \\
    }= \pmatrix{
   -6\\
6 \\
3 \\
7 \\ 
 }$$
Here are my calculations:
$$\left( \begin{array}{ccc|c}
    -1 &  -2& -3 & -6 \\
    3 &  2& 1 & 6 \\
    1 &  1& 1 & 3 \\
    2 &  4& 1 & 7 \\    
    \end{array} \right)$$
row4 = row1*2 + row4 &&
row2 = row3*3 -row2
$$\left( \begin{array}{ccc|c}
    -1 &  -2& -3 & -6 \\
    0 &  -1& 2 & -3 \\
    1 &  1& 1 & 3 \\
    0 &  0& -5 & -5 \\    
    \end{array} \right)$$
row4 = row4/-5 &&
row1 = row3 + row1
$$\left( \begin{array}{ccc|c}
    0 &  -1& -2 & -3 \\
    0 &  -1& 2 & -3 \\
    1 &  1& 1 & 3 \\
    0 &  0& 1 & 1 \\    
    \end{array} \right)$$
row1 = row4*4 + row1 
$$\left( \begin{array}{ccc|c}
    1 &  1& -1 & 3 \\
    0 &  -1& 2 & -3 \\
    0 &  0& 1 & 1 \\
    0 &  0& 0 & 1 \\    
    \end{array} \right)$$
I get as a solution an unsolvable matrix; however, the matrix should be solvable. What was wrong with my calulations? I cannot see the problem.
Thx in advance!!!

In the third matrix row2=row3*3-row2,  $-1$ must be $1$ and $-3$ must be $3$. Thus the second row is $[0,~ 1,~ 2| ~3]$
