Is a Matrix Invertible in Modular Arithmetic if det(A) and n are Coprime?

  • Context: Undergrad 
  • Thread starter Thread starter nahuel_pelado
  • Start date Start date
  • Tags Tags
    Arithmetic Matrices
Click For Summary
SUMMARY

A matrix is invertible in modular arithmetic modulo Zn if its determinant, det(A), is coprime with n, meaning gcd(det(A), n) = 1. This condition ensures that the matrix has an inverse in the modular system. Additionally, the verification of the inverse can be performed using the equation A.A^-1 mod n = A^-1.A mod n = I, where I represents the identity matrix. Understanding unimodular matrices, where the determinant is a non-zero constant, further clarifies the relationship between invertibility and determinants.

PREREQUISITES
  • Understanding of modular arithmetic, specifically Zn.
  • Knowledge of determinants and their properties in linear algebra.
  • Familiarity with the concept of coprimality and gcd (greatest common divisor).
  • Basic understanding of unimodular matrices and their characteristics.
NEXT STEPS
  • Research the properties of modular arithmetic and its applications in linear algebra.
  • Study the concept of determinants in depth, focusing on their role in matrix invertibility.
  • Learn about unimodular matrices and their significance in polynomial algebra.
  • Explore examples of matrices in Zn to practice determining invertibility based on the determinant.
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, modular arithmetic, and matrix theory, will benefit from this discussion.

nahuel_pelado
Messages
2
Reaction score
0
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:
Physics news on Phys.org
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.
 
Why does a real matrix with invertible determinant have to have an inverse?
 

Similar threads

MHB Creating Linearly Dependent Square Matrix from Cam Mechanism Equation
  • · Replies 2 ·
Replies
2
Views
2K
MHB Is B Equal to A³ Given Symmetric and Invertible Matrices?
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Open Conditions for Matrix Groups: Understanding the Role of det(A)≠0
  • · Replies 6 ·
Replies
6
Views
2K
Graduate Structure of modulo-integer matrix group GL(r,Z(n))?
  • · Replies 17 ·
Replies
17
Views
7K
Undergrad Proving only 1 normalized unitary vector for normal matrix
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Simple Linear Algebra (determinant invertibility)
  • · Replies 6 ·
Replies
6
Views
6K
Undergrad How to count the total # of non-invertible 2x2 matrices
  • · Replies 7 ·
Replies
7
Views
7K
Undergrad Dfns group action & group, written in terms of the other
  • · Replies 26 ·
Replies
26
Views
909
Prove the identity matrix is unique
  • · Replies 69 ·
3
Replies
69
Views
9K
Is SU(n) a Group Under Matrix Multiplication?
  • · Replies 5 ·
Replies
5
Views
2K