Question

The number of 3x3 orthogonal matrices with integer entries is

Answer 1

Answer:

Let A be such a matrix, and a1, a2 and a3 its columns. Given that A is orthogonal means that

⟨ai,aj⟩={10 if i=j if i≠j

So in particular ⟨ai,ai⟩=1. Because the ai have all integer coefficients, it follows that

a1,a2,a3∈{(1,0,0),(−1,0,0),(0,1,0),(0,−1,0),(0,0,1),(0,0,−1)},

leaving six options for each of a1, a2 and a3. For any given value of a1 there are precisely 4 values of a2 such that ⟨a1,a2⟩=0. For any given values of a1 and a2 there are precisely 2 values of a3 such that ⟨a1,a3⟩=⟨a2,a3⟩=0. Hence there are 6×4×2=48 such matrices in total.

i hope this will help uh