eigenvalues by inspection

Well, May be my question does not make any sense, But one of my junior asked me whether we can say eigenvalues of a matrix by inspection may be for $3\times 3$ matrix? He said for the following matrix, will $3$ be an eigenvalue with multiplicity $3$ with out calculation? $$\left( \begin{array}{ccc} 3 & 2 & 2 \\ 2 & 3 & 2 \\ 2 & 2 & 3 \\ \end{array} \right).$$ So far I know that if it is real symmetric then its eigenvalues are purely real, if skew then purely imaginary, if diagonal then all the entries, and if unitary then with modulus 1, known facts at all.If any one knows more about this please write.

In this case you can see the eigenvalues "by inspection".

Item #1: If you subtract the identity matrix from your matrix, you get three repeated rows, i.e. $A-I$ has rank $1$ only. This means that the eigenspace associated with $\lambda=1$ is $2$-dimensional, IOW it is a double eigenvalue (and a double root of characteristic polynomial).

Item #2: The sum of entries on all rows is equal to $7$. This means that the vector $(1,1,1)^T$ is an eigenvector belonging to $\lambda=7$.

This is a 3x3 matrix, so that's all.

The Gershgorin circle theorem comes close to estimating the eigenvalues by 'inspection' - by summing the absolute values of the row elements (except the ones on the diagonal). So in this case the three eigenvalues are all in the interval [3-4,3+4].