Question 36
A nonempty finite set $S$ of complex numbers has the property: $xy \in S$, for every $x, y \in S$.
a) Prove that, for every $x \in S$, $\frac{1}{x} \in S$.
b) Let $a, b$ be two fixed elements of $S$ and $m, n \in \mathbb{N}$, $m, n \ge 2$. Find the number of $m \times n$ matrices, with elements from $S$, having the product of the elements of each line equal to $a$ and the product of the elements of each column equal to $b$.
a) Prove that, for every $x \in S$, $\frac{1}{x} \in S$.
b) Let $a, b$ be two fixed elements of $S$ and $m, n \in \mathbb{N}$, $m, n \ge 2$. Find the number of $m \times n$ matrices, with elements from $S$, having the product of the elements of each line equal to $a$ and the product of the elements of each column equal to $b$.