Let $a_1,\dots,a_n$ be real numbers, and set $a_{ij} = a_ia_j$. Consider the $n \times n$ matrix $A=(a_{ij})$. Then
It is possible to choose $a_1.\dots,a_n$ such that $A$ is non-singular
matrix $A$ is positive definite if $(a_1,\dots,a_n)$ is nonzero vector
matrix $A$ is positive semi definite for all $(a_1,\dots,a_n)$
for all $(a_1,\dots,a_n)$, $0$ is an eigen value of $A$
Please help. I can't even make a guess.