[Math] Sufficient conditions for positive semidefinite matrices

Question

• Is a symmetric matrix with positive terms (i.e., $a_{ij} > 0$) and positive determinant positive semidefinite?

• Is a symmetric matrix with positive terms, positive determinant, and
  terms that satisfy $a_{ij}^2 \leq a_{ii} a_{jj}$ positive semidefinite?

Answer 1

The answer to your first question is negative. Consider$$A=\begin{pmatrix}1&2&2\\2&2&2\\2&2&1\end{pmatrix}.$$Each entry is greater than $0$ and the determinant is $2$. However$$\begin{pmatrix}-1&0&1\end{pmatrix}A\begin{pmatrix}-1\\0\\1\end{pmatrix}=-2.$$