Positive definite and Negative Definite

Question

Let $a_{ij} = a_ia_j, 1\le i, j\le b$, where $a_1,.......,a_n$ are real numbers. Let $A= ((a_{ij}))$ be the n*n matrix $(a_{ij})$. Then

1. It is possible to choose a_1,.........,a_n so as to make the matrix A non singular.
2. The matrix A is positive definite if (a_1,........,a_n) is a non zero vector.
3. The matrix A is positive semi definite for all (a_1,....,a_n).
4. For all (a_1,.....,a_n),zero is an eigen value of A.

Answers are 1 and 2 but I could not understand why! Please explain.