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