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 $3$ 

$\begin{bmatrix} 5&amp;1&amp;1&amp;1&amp;1&amp;1\\1&amp;5&amp;1&amp;1&amp;1&amp;1\\1&amp;1&amp;5&amp;1&amp;1&amp;1\\1&amp;1&amp;1&amp;4&amp;0&amp;1\\1&amp;1&amp;1&amp;0&amp;3&amp;0\\1&amp;1&amp;1&amp;1&amp;0&amp;4\end{bmatrix}$.

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.
Call the matrix $A$. It is real symmetric, hence diagonalisable. Therefore, it suffices to show that the rank deficiency of $A-4I$ is $3$.
Hint:$A=\begin{bmatrix}
4 &amp; 0 &amp; 0 &amp;0&amp;0&amp;0\\
0&amp;4&amp;0&amp;0&amp;0&amp;0\\
0&amp;0&amp;4&amp;0&amp;0&amp;0\\
0&amp;0&amp;0&amp;3&amp;0&amp;0\\
0&amp;0&amp;0&amp;0&amp;2&amp;0\\
0&amp;0&amp;0&amp;0&amp;0&amp;3
\end{bmatrix}+\begin{bmatrix}
1 &amp; 1 &amp; 1 &amp;1&amp;1&amp;1\\
1 &amp; 1 &amp; 1 &amp;1&amp;1&amp;1\\
1 &amp; 1 &amp; 1 &amp;1&amp;1&amp;1\\
1 &amp; 1 &amp; 1 &amp;1&amp;1&amp;1\\
1 &amp; 1 &amp; 1 &amp;1&amp;1&amp;1\\
1 &amp; 1 &amp; 1 &amp;1&amp;1&amp;1
\end{bmatrix}$
$$\text{Both matrices commute, eigenvalues for the first are} ~~4,4,4,3,2,3$$
$$\text{eigenvalues for thesecond matrix are}~~0,0,0,0,0,6$$