Answer
\begin{pmatrix} 1 & 0 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}
Work Step by Step
For $\begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix} $ to be a solution, we can have $x_1 = x_3$ and $x_2$ to be a free variable. The matrix which matches these conditions is:
\begin{pmatrix} 1 & 0 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}
Checking that $\begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix} $ is a solution, we find:
\begin{equation}
\begin{pmatrix} 1 & 0 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} * \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}
\end{equation}
Thus, the matrix satisfied all of the problem conditions.
