# Modular arithmetic in matrices

1. Jul 15, 2010

This is the question: What must fulfill a matrix to be invertible in module Zn? Demonstrate. Z refers to integers.

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?: A.A^-1 mod n = A^-1.A mod n = I ... I = identity matrix

Last edited: Jul 16, 2010
2. Jul 16, 2010

### trambolin

Hint : What is a unimodular matrix?

Another hint: A square polynomial matrix U in $\mathbb{R}^{n\times n}[x]$ is unimodular if the determinant $\det U \neq 0$ is a constant. The inverse of a unimodular polynomial matrix is again a polynomial matrix.

3. Jul 16, 2010

### Hurkyl

Staff Emeritus
Why does a real matrix with invertible determinant have to have an inverse?