Question 10
Let $x_{1}, \ldots, x_{n}$ and $y_{1}, \ldots, y_{n}$ be real numbers. Let $A = (a_{ij})_{1 \leq i,j \leq n}$ be the matrix with entries
$$ a_{ij}=\begin{cases}1,&\text{if } x_{i}+y_{j}\geq0;\\0,&\text{if } x_{i}+y_{j} \lt 0.\end{cases} $$
Suppose that $B$ is an $n \times n$ matrix with entries 0, 1 such that the sum of the elements in each row and each column of $B$ is equal to the corresponding sum for the matrix $A$. Prove that $A = B$.
$$ a_{ij}=\begin{cases}1,&\text{if } x_{i}+y_{j}\geq0;\\0,&\text{if } x_{i}+y_{j} \lt 0.\end{cases} $$
Suppose that $B$ is an $n \times n$ matrix with entries 0, 1 such that the sum of the elements in each row and each column of $B$ is equal to the corresponding sum for the matrix $A$. Prove that $A = B$.