Let $n$ be an integer and $n_1$ be one of its divisors. Let $A$ be a $n \times n$ symmetric matrix defined by $a_{i,i} = 4$, $a_{i,i+1} = a_{i+1,i} = -1$ for all $i$ such that $1 \le i \le n-1$ and $i+1$ is not a multiple of $n_1$, $a_{i,i+n_1} = a_{i+n_1,i} = -1$ and $a_{i,j} = 0$ otherwise.