Difficult# Dependence and Column Vectors

LINALG-JVECWQ

Let $A$ be a $n\times n$ square matrix with column vectors $v_1,v_2,\ldots,v_n$.

Suppose $a_1,a_2,\ldots,a_n$ are real numbers such that:

$$a_1v_1+a_2v_2+\ldots+a_nv_n=0$$

Suppose further that the product $a_1a_2\cdots a_n\neq0$.

Given just this information, what is the maximum possible rank of $A$?