<s>[INST] Show that the following matrix has $4$ as an eigen value of multiplicity $3$ Show that the following matrix has $4$ as an eigen value of multiplicity $3$ 
$\begin{bmatrix} 5&1&1&1&1&1\\1&5&1&1&1&1\\1&1&5&1&1&1\\1&1&1&4&0&1\\1&1&1&0&3&0\\1&1&1&1&0&4\end{bmatrix}$.
How should I try this?Isn't this equivalent to finding the entire spectrum of the above matrix?
Or there are some tricks involved here?Please help.
 [/INST] Hint:$A=\begin{bmatrix}
4 & 0 & 0 &0&0&0\\
0&4&0&0&0&0\\
0&0&4&0&0&0\\
0&0&0&3&0&0\\
0&0&0&0&2&0\\
0&0&0&0&0&3
\end{bmatrix}+\begin{bmatrix}
1 & 1 & 1 &1&1&1\\
1 & 1 & 1 &1&1&1\\
1 & 1 & 1 &1&1&1\\
1 & 1 & 1 &1&1&1\\
1 & 1 & 1 &1&1&1\\
1 & 1 & 1 &1&1&1
\end{bmatrix}$
$$\text{Both matrices commute, eigenvalues for the first are} ~~4,4,4,3,2,3$$
$$\text{eigenvalues for thesecond matrix are}~~0,0,0,0,0,6$$
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