How to find eigen values of such kind of matrices ( Matrices which are bigger than $ 3\times 3$) The following question is part of a masters exam for which I am preparing and I don't have any methodology on how this type of matrices are solved.
This was question asked:
I don't know how to find eigenvalues of matrices ( larger than 3 $\times$ 3).
So, it's my humble request if you can tell some details or source of results and observations which are useful for dealing with such matrices.
I have studied linear algebra from David C lay.
Thanks a lot!!

In addition to the existing answers I'd like to say that it's not hard to calculate the characteristic polynomial using Laplace expansion:
\begin{align}p(x)&=\det\begin{bmatrix} x & 0 & 0 & -1 & 0 & 0\\
0 & x & 0 & 0 & -1 & 0\\
0 & 0& x&  0 & 0& -1\\
-1 & 0 &0&x& 0 & 0\\
0 & -1 & 0 & 0&x &0\\
0 & 0& -1 & 0 & 0 &x 
 \end{bmatrix}\\
&=x\det\begin{bmatrix}
 x & 0 & 0 & -1 & 0\\
 0& x&  0 & 0& -1\\
 0 &0&x& 0 & 0\\
 -1 & 0 & 0&x &0\\
 0& -1 & 0 & 0 &x 
 \end{bmatrix}+1\det\begin{bmatrix} 0 & 0 & -1 & 0 & 0\\
 x & 0 & 0 & -1 & 0\\
 0& x&  0 & 0& -1\\
 -1 & 0 & 0&x &0\\
 0& -1 & 0 & 0 &x 
 \end{bmatrix}\\
&=x\cdot x\det\begin{bmatrix}
 x & 0  & -1 & 0\\
 0& x&  0 & -1\\
 -1 & 0 &x &0\\
 0& -1 &  0 &x 
 \end{bmatrix}-1\det\begin{bmatrix}
 x & 0  & -1 & 0\\
 0& x & 0& -1\\
 -1 & 0 &x &0\\
 0& -1  & 0 &x 
 \end{bmatrix}\\
&=(x^2-1)\det\underbrace{\begin{bmatrix}
 x & 0  & -1 & 0\\
 0& x & 0& -1\\
 -1 & 0 &x &0\\
 0& -1  & 0 &x 
 \end{bmatrix}}_{=M}
\end{align}
Now it's your turn to prove $\det(M)=(x^2-1)^2$ using the same method (Laplace expansion) as above.
The final result is

$p(x)=(x^2-1)^3$

