Let $a_{i,j} =a_ia_j$ , $1 ≤ i, j ≤ b$, where $(a_1,a_2,\ldots,a_n)$, are real numbers. Let $A =(a_{i,j})$ be the $n ×n$ matrix.
Then
It is possible to choose $(a_1,a_2.,….,a_n)$, so as to make the matrix A non singular
The matrix $A$ is positive definite if $(a_1,a_2,\ldots,a_n)$, is a non zero vector
The matrix $A$ is positive semi definite for all $(a_1,a_2,\ldots,a_n)$,
For all $(a_1,a_2,\ldots,a_n)$, zero is an eigenvalue of $A$.
For second order matrix I checked that (1) and (2) are not correct. But I am not sure about the others that they are true/false. Thanks to help me.