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.

  1. Prove that the order of $H$ is $2^nn!$.
  • $\begingroup$ Remember that orthogonal matrices represent isometries, and preserve orthonormal bases. The column vectors are the respective images of the standard basis vectors. So, you want an orthonormal basis with integer coefficients; there aren't many of those! $\endgroup$ – Theo Bendit Mar 12 '18 at 4:02
  • $\begingroup$ Ok, so I know that orthogonal matrices represent reflection or rotation, but what do you mean they preserve orthonormal bases? $\endgroup$ – user539807 Mar 12 '18 at 4:17
  • $\begingroup$ If you have an orthonormal basis of column vectors $(e_1, \ldots, e_n)$ and $M \in H$, then $(Me_1, \ldots, Me_n)$ is another orthonormal basis of column vectors. Have you done anything with orthonormal bases yet? $\endgroup$ – Theo Bendit Mar 12 '18 at 4:21
  • $\begingroup$ Not much. I know that orthonormal basis vectors should have length 1 and should be linearly independent. $\endgroup$ – user539807 Mar 12 '18 at 4:27
  • 1
    $\begingroup$ As I said in the first comment, orthonormal bases with integer coordinates are very rare! $\endgroup$ – Theo Bendit Mar 12 '18 at 4:51

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