<s>[INST] Elementary matrix operations I am taking an introductory/first year course in Linear Algebra and I am at my wits' end with the following problem.
I am asked to find both the nullspace and the general solution of the following systems $A\tilde{x}=\tilde{b}$.
The first is
$$
A=
\begin{pmatrix}
1 & 1 & 0 & 2\\
2 & 1 & 1 &-2\\
2 & 2 & 2 & 1
\end{pmatrix},\quad \tilde{b}=
\begin{pmatrix}
12\\0\\14
\end{pmatrix}
$$
Proceeding with an augmented matrix,
I have
\begin{align*}
\left(
\begin{array}{rrrr|c}
1 & 1 & 0 & 2 & 12\\
2 & 1 & 1 &-2 & 0\\
2 & 2 & 2 & 1 & 14
\end{array}
\right)&\sim
\left(
\begin{array}{rrrr|r}
1 & 1 & 0 & 2 & 12\\
0 &-1 & 1 &-6 & -24\\
0 & 0 & 2 & -3 &-10
\end{array}
\right)\\
&\sim
\left(
\begin{array}{rrrr|r}
1 & 1 & 0 & 2 & 12\\
0 & 1 &-1 & 6 & 24\\
0 & 0 & 1 & -3/2 &-5
\end{array}
\right)
\end{align*}
I've checked this multiple times and can't find a mistake, plus it agrees with the solutions my professor has provided.
Solving this now, I have
\begin{align*}
x_4&=\alpha\in\mathbb{R}\\
x_3&=-5+\frac{3}{2}\alpha\\
x_2&=19-\frac{9}{2}\alpha\\
x_1&=-7+\frac{5}{2}\alpha
\end{align*}
So from this, my solution will be
$$
\tilde{x}=
\begin{pmatrix}
-7+\frac{5}{2}\alpha\\
19-\frac{9}{2}\alpha\\
-5+\frac{3}{2}\alpha\\
\alpha
\end{pmatrix}
=
\begin{pmatrix}
-7\\19\\-5\\0
\end{pmatrix}+\beta
\begin{pmatrix}
5\\-9\\3\\2
\end{pmatrix},\quad\beta\in\mathbb{R}.
$$
This does not agree with the solutions given, and the solutions given are just solutions without any steps in between so I'm a little confused as to where I went wrong. The weird part is, my nullspace agrees with the solution provided but the particular solution certainly does not.
Edit: My professor's solution is
$$
\tilde{x}=
\begin{pmatrix}
3\\1\\1\\4
\end{pmatrix}+\alpha
\begin{pmatrix}
5\\-9\\3\\2
\end{pmatrix},\quad\beta\in\mathbb{R}.$$
Can somebody please let me know what I've done incorrectly?
Thank you.
 [/INST] Your solution is correct.  You can obtain the professor's solution from yours via the substitution $\beta=\alpha+2$. Both are valid answers, and unless there was some given format your answer was supposed to be in, I don't see why yours could be considered incorrect.
</s>