Hello! I have just one question!

We have the ring $\displaystyle M_{2}(\mathbb{Z}_{3})$, which are all 2x2 matrices over the field $\displaystyle \mathbb{Z}_{3}$. We need to find all the elements of ring $\displaystyle K^{*}$ (the set of multiplicative inverses).

I know that $\displaystyle A \in K^{*}$ if the $\displaystyle det(A) \neq 0 $. But here in $\displaystyle \mathbb{Z}_{3}$ we just have $\displaystyle (det(A) = 1)$ or $\displaystyle (det(A) = 2)$.