Let $A$ be a real symmetric matrix. Show that all the eigenvalues of $A$ are real, and that the eigenvectors corresponding to distinct eigenvalues are orthogonal to each other.

Find the eigenvalues and eigenvectors of

$$A=\left(\begin{array}{ccc} 2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2 \end{array}\right)$$

Give an example of a non-zero complex $(2 \times 2)$ symmetric matrix whose only eigenvalues are zero. Is it diagonalisable?