linear algebra - Find the cardinality of the set of permutation matrices.

Let $S$ be the set of all $3\times3$ matrices $A$ with integer entries such that $AA^t=I$, where $A^t$ is the transpose of $A$. Then $|S|=?$

1. $12$


2. $24$


3. $48$


4. $60$

I tried to find the structure of such matrices from an arbitrary matrix and using $AA^t=I$ and found that these are permutation matrices. So how to calculate the cardinality of set of permutation matrices? Isn't it $n!$? But that does not give me anything to relate with the options given. Any hint or help please.

Answer

Certainly permutation matrices satisfy $AA^\top=I$, but there are others! For example,
$$

A=
\left[\begin{array}{rrr}
-1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]
$$

The condition $AA^\top=I$ forces the columns of $A$ to be orthonormal. Since $A$ is integral, it follows that each column of $A$ has exactly one nonzero entry, which must be $\pm1$. Moreover, the corresponding entries in the other rows must be zero, to ensure orthogonality.

There are three places to place $\pm1$ in the first column, two places in the second, and one in the third. Hence the number of integral matrices satisfying $AA^\top=I$ is
$$
(2\cdot 3)(2\cdot 2)(2\cdot 1)=48
$$