Let $S$ be the set of all $3 \times 3$ matrices $A$ with integer entries such that the product $AA^t$ is the identity matrix.
Then $|S|$ is :
23.
24
48
60
Answer is $48$ but I don't know how to solve it.
> Note that each column of $A$ must be an integer vector of unit length which means that each column is of the form $\pm e_i$ for some $1 \leq i \leq 3$ (where $e_i$ are the standard basis vectors).
Well, so if you have a vector $(a,b,c)$ where all coordinates are integers, then you get $$a^2+b^2+c^2=1.$$
If you add a condition that a,b,c have to be integers, what possibilities do you have for them?
> Thus, we need to pick a permutation of the $e_i$'s to put as columns and then, for each column independently, decide whether it gets a plus or a minus sign.
You agree that you see that columns can only be $\pm e_1$, $\pm e_2$, $\pm e_3$.
So it is clear that if we ignore signs then the columns are some permutation of $e_1$, $e_2$, $e_3$. (Simply because we cannot have any of them twice.)
Well, orthogonal vectors are automatically linearly independent.
BTW good catch that there is a duplicate. I have added a comment below yours linking to this chat. (So that people seeing your comment know that you already have an answer.)