Are $2 × 2$ minors sufficient for positive definiteness?

Question

Positive definiteness implies positive determinants of all $2 × 2$ minors. I'm curious about a converse statement:

Suppose $|m_{ij}| < \sqrt{m_{ii}m_{jj}}$ for all $i,j$ and $m_{ii} > 0$. Is it possible for $m_{ij}$ to not be positive definite?

Answer 1

Consider the $3 \times 3$ matrix with all $1$s on the diagonal and all $-0.9$ elsewhere.