For a positive integer $n$ and a square matrix $A$ with integral entries, let $\text{ord}(A, n)$ be the smallest positive integer $k$ such that $A^k \equiv \mathbf{1} \bmod n$, if such integer $k$ exists, otherwise put $\text{ord}(A, n) := +\infty$. So $\text{ord}(A, n)$ is the *order of $A$ modulo $n$*.

We have the following result [1, Theorem 17].

**Theorem.** If $\delta > 0$ and $A$ is $2 \times 2$ matrix with determinant equal to $1$ and trace not less than $2$, then
$$\text{ord}(A,n) \gg n^{1/2} \exp((\log n)^\delta),$$
for all positive integers $n$ but a set of asymptotic density $0$.

*My question is:* Are there other results of this kind for square matrices with arbitrary sizes and determinants? Are there stronger results?

Thanks.

[1] *P. Kurlberg and Z. Rudnick, On Quantum Ergodicity for Linear Maps of the Torus, Commun. Math. Phys. (2001) 222, 201-227.*