<s>[INST] How to find the eigen values of the given matrix 
Given the matrix 
\begin{bmatrix}
5&1&1&1&1&1\\1&5&1&1&1&1\\1&1&5&1&1&1\\1&1&1&4&1&0\\1&1&1&1&4&0\\1&1&1&0&0&3
\end{bmatrix}
find its eigen values(preferably by elementary row/column operations).

Since I don't know any other method other than elementary operations to find eigen values so I tried writing the characteristic polynomial of the matrix  which is follows:

\begin{bmatrix}
x-5&-1&-1&-1&-1&-1\\-1&x-5&-1&-1&-1&-1\\-1&-1&x-5&-1&-1&-1\\-1&-1&-1&x-4&-1&0\\-1&-1&-1&-1&x-4&0\\-1&-1&-1&0&0&x-3
\end{bmatrix}

Using $R1=R1-(R2+R3+R4+R5+R6)$

\begin{bmatrix}
x&-x+8&-x+8&-x+6&-x+6&-x+4\\-1&x-5&-1&-1&-1&-1\\-1&-1&x-5&-1&-1&-1\\-1&-1&-1&x-4&-1&0\\-1&-1&-1&-1&x-4&0\\-1&-1&-1&0&0&x-3
\end{bmatrix}

 [/INST] The characteristic polynomial $\chi_t$ of this matrix can be factored nicely, i.e.,
$$
\chi_t(A)=(t - 2)(t - 3)(t - 4)^3(t - 9).
$$
So the six eigenvalues are $2,3,4,4,4,9$. 
</s>