<s>[INST] Determine the unique vector x in the row space for A, for which Ax = b Given the Vector $b = (9, -6, 3)^T$
$A =\begin{pmatrix}
         1 & 2 & 0 & 1\\
         -1 & -1 & -1 & 0 \\
         1 & 0 & 2 &-1
        \end{pmatrix}$
I know that the row space is $ \begin{pmatrix} 1 & 0 & 2& -1\end{pmatrix}$ , $ \begin{pmatrix} 0 & 1 & -1& 1\end{pmatrix}$
Not really sure how to proceed.
 [/INST] As far as I can see, this system of equations doesn't even a have a solution, for we have:
$$
\begin{align}
\left[
\begin{array}{cccc|c}
1&2&0&1&6\\
-1&-1&-1&0&-6\\
1&0&2&-1&3
\end{array}
\right]
&\iff
\left[
\begin{array}{cccc|c}
1&2&0&1&6\\
0&1&-1&1&0\\
0&-2&2&-2&-3
\end{array}
\right]
\iff
\left[
\begin{array}{cccc|c}
1&2&0&1&6\\
0&1&-1&1&0\\
0&0&0&0&-3
\end{array}
\right]
\end{align}
$$
</s>