$\begin{align} & \left[ \begin{array}{cccc|c}
1 & 1 & 0 & \lambda & 1 \\
\lambda & -1 & 1 & 0 & 1 \\
\lambda & 3 & -1 & 2 & 1
\end{array} \right] \sim \left[ \begin{array}{cccc|c}
1 & 1 & 0 & \lambda & 1 \\
\lambda & -1 & 1 & 0 & 1 \\
2 \lambda & 2 & 0 & 2 & 2
\end{array} \right] \sim \left[ \begin{array}{cccc|c}
1 & 1 & 0 & \lambda & 1 \\
\lambda & -1 & 1 & 0 & 1 \\
\lambda & 1 & 0 & 1 & 1
\end{array} \right] \\ \sim & \left[ \begin{array}{cccc|c}
1 & 1 & 0 & \lambda & 1 \\
\lambda + 1 & 0 & 1 & \lambda & 2 \\
\lambda - 1 & 0 & 0 & -\lambda + 1 & 0
\end{array} \right]
\end{align}$