Consider the map: $\varphi:GL(2,\mathbb{Z}_3)\rightarrow\mathbb{Z_3^*}$ with the function being $\varphi(M)=\det(M)$. I've shown that the function is a group homomorphism, and next I have to calculate the kernel and image.
I know the kernel is all the elements which maps to the second identity, written as:
$$\ker(\varphi)=\{M\in GL(2,\mathbb{Z}_3)\mid\det(\varphi)=1\}$$
But how do I do this? I have $3^4=81$ possible 2x2 matricies to check - I've gone through a lot of them, but is there an easier way to write this, than writing every single one? Because if this was an exam question, I think I would use a lot of time :(
The same goes for the image.
Thank you in advance