Let $H$ be the set of matrices in $O_n(\mathbb{R})$ whose matrix entries are integers.

Question

1. Prove that $H$ is equal to the set of all $n\times n$ matrices $A=(a_{ij})$ with integer coefficients such that $\vert a_{ij}\vert=0$ or $1$ for all $i,j$ and each row and column of $A$ has one non-zero entry.

So every row and column of $A$ has exactly one non-zero entry, which is either $1$ or $-1$. I don't really know where to start with this, and presumably the next part of the question is also link with this.

2. Prove that the order of $H$ is $2^nn!$.