Are symmetric binary matrices necessarily positive semi-definite?

Let $A$ be a symmetric $n\times n$ matrix with entries only 0 or 1 and the diagonal entries of $A$ are all 1. Is A positive (semi-) definite?

$$\begin{pmatrix}1&0&1\\0&1&1\\1&1&1\end{pmatrix}$$
• The trace is $3$, the determinant is $-1$. – Git Gud Feb 17 '15 at 10:16