compute rank of $2\times n$ matrix Let's consider matrix
$$
A = 
\left [
\begin{array}{cccc}
a_1 & a_2 & \ldots & a_n\\
b_1 & b_2 & \ldots & b_n\\
\end{array}
\right ]
$$
where $(a_i,b_i)\in\mathbb{N}^2\setminus\{(0,0)\}$ $i=1,2,\ldots,n$ ($a_i$ or $b_i$ could be equal to zero but not both of them), let's introduce notation for $i,j=1,2,\ldots,n$, $i<j$:
$$
A_{ij} = 
\left |
\begin{array}{cc}
a_i & a_j\\
b_i & b_j\\
\end{array}
\right |
$$
My questions:
a) I know that:
$$
\begin{array}{l}
A_{12} + A_{13} +\ldots + A_{1n} = 0\\
\sum\limits_{i=1}^{j-1}A_{ij} - \sum\limits_{i=j+1}^{n}A_{ji} = 0\ \ \ \text{for $j=2,3,\ldots,n$}
\end{array}
$$
Does from this follow that $\text{rank}\;A = 1$ ?
b) We have $n$ linear equations such that in a) (they are linear independent). Does from this follow that $\text{rank}\;A = 1$ ?
c) There are $\frac{n(n-1)}{2}$ $A_{ij}$s. How many linear equations (let's denote this number by $k$) such that in a), b) should I write to be sure that  $\text{rank}\;A = 1$? (of course I want to know minimal number $k$)
 A: The matrix $A$ is of rank $0$ if and only if the two rows are zero, but you eliminated that possibility explicitly. $A$ is of rank at most $1$ if and only if all the $A_{ij}$ are $0$. Among the $A_{ij}$ there are linear dependencies such as $A_{ij}b_k-A_{ik}b_j+A_{jk}b_i=0$ and the same with $b$, thus $n-1$ are linearly independent.
For your question in (a), since $A_{ij}=-A_{ji}$, let $s_a:=\sum_ia_i,$ $s_b:=\sum_ib_i,$ and $e_i:=\sum_jA_{ij}$ where
$e_i=a_is_b-b_is_a.$ Each equation is $e_i=0$ and the sum of all of these equations is $0$, and they are enough to ensure rank $1$. For your question in (b) any $n-1$ linearly independent equations such as in (a) is enough to ensure rank $1$. Same thing for question in (c). The minimal number is $n-1$.
This situation has an interpretation in projective geometry. We have $n$ points $P_i:=[a_i,b_i]$ given in homogeneous coordinates on a projective line. The $A_{ij}=0$ is the condition that $P_i=P_j$. There are only $n-1$ independent such conditions since each independent condition reduces the number of distinct points by $1$ and the minimum number of distinct points is $1$. 
