Modular arithmetic in matrices

[nahuel_pelado]

This is the question: What must fulfill a matrix to be invertible in module [tex]\mathbb{Z}n[/tex]? Demonstrate. Z refers to integers. n could be any number, for example if the module would be [tex]\mathbb{Z}_5[/tex] all the numbers in the matrix must be between 0 and 4.

I really appreciate that someone could help me with this because i couldn't find strong information about it.
I think that considering A as a matrix... the det(A) must be coprime with the module (n), so that gcd(det(A),n)=1 but i'm not sure about it.

In case that a matrix has inverse in module Zn, is correct to use this to verify?: [tex]A\,.[/tex][tex]A^{-1}[/tex] [tex]mod\, n =[/tex][tex]A^{-1}[/tex][tex].\,A\, mod\, n = I[/tex] ´´´´´´´´ I = identity matrix