For $n=2$, the growth of $S(r)$ is actually significantly slower than i707107's bounds would suggest. I will show that $A_t = O(t r^2)$, not $t r^3$, and hence $S(r) = O(r^2 \log r)$. I can also establish the lower bound $S(r) > c r^2$, so $2$ is the correct exponent.

Let $A(r,t)$ be the number of $2 \times 2$ matrices with $||A|| \leq r$ and $|\det A| \leq t$, $\det A \neq 0$. We will show $A(r,t) \leq C r^2 t$ and $A(r,1) \geq c r^2$ for some constants $c$ and $C$.

**The upper bound:** We will count the number of matrices $\left( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right)$ with $0 < |\det A| \leq t$, $||A|| \leq r$ and $|d| = \max(|a|,|b|,|c|,|d|)$. Multiplying by $4$ then gives an upper bound on $A(r,t)$. We break up our count according to $GCD(c,d)$.

First, let's count the terms with $GCD(c,d)=1$. Fix a particular $(c,d)$. For each $u$ between $-t$ and $t$, the equation $ad-bc=u$ determines $b \bmod d$. Combined with the condition $|b| \leq |d|$, there are only two values for $b$, given a fixed $(c,d)$ and a fixed value for $\det \left( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right)$. And, $(b,c,d,u)$ determines $a$. There are $O(r^2)$ choices for $(c,d)$, and $O(t)$ choices for $u$, so such terms contribute $O(t r^2)$ to our count.

Now, let's do the case where $GCD(c,d) = g$, say $c = g c'$ and $d = g d'$. The equation $ad-bc=u$ is only solvable at all when $g|u$, which is for $O(t/g)$ values of $u$. In that case, $b$ is determined modulo $d'$. Combined with the inequality $|b|\leq d$, this gives $O(d/d') = O(g)$ choices for $b$. So, for each $(c,d)$, there are $O(g) O(t/g) = O(t)$ ways to complete it to a $(a,b,c,d,u)$ quintuple. The number of $(c,d)$ pairs with $GCD$ equal to $g$ is $O(r^2/g^2)$.

So our bound is $O \left( \sum_g t r^2/g^2 \right) = O(t r^2)$.

**The lower bound** There are $(6/\pi^2 - o(1)) r^2$ pairs $(c,d)$ with $GCD(c,d)=1$ and $|c|, |d| \leq r^2$. For each of these, we can find $(a,b)$ with $|a|$, $|ab| \leq \max(|c|, |d|)$ and $ad-bc=1$. So there are $(6/\pi^2 - o(1)) r^2$ matrices contributing to $A(r,1)$. (It is easy to improve this constant, but probably not worthwhile.)