\sim-\left|
\begin{array}{cccc}
 -1 & 1 & 1 & -\lambda  \\
 0  & 1-\lambda  & 0 & 1-\lambda  \\
 0 & 0 & 1-\lambda  & 1-\lambda  \\
 0 & 0 & 1-\lambda  & \lambda ^2+\lambda -2
\end{array}
\right|
\begin{array}{c}
 
\begin{array}{c}
 -\text{III}_v+\text{IV}_v \\
 \sim 
\end{array}
 \\
 \square 
\end{array}
-\left|\begin{array}{cccc}
 -1 & 1 & 1 & -\lambda  \\
 0  & 1-\lambda  & 0 & 1-\lambda  \\
 0 & 0 & 1-\lambda  & 1-\lambda  \\
 0 & 0 & 0 & \lambda ^2+2\lambda -3
\end{array}
\right|=(1-\lambda )^2\left(\lambda ^2+2\lambda -3\right).