Modular arithmetic in matrices

In summary, for a matrix to be invertible in module Zn, its determinant must be coprime with the module (n), meaning that their greatest common divisor is 1. This can be verified by checking if A.A^-1 mod n = A^-1.A mod n = I, where I is the identity matrix. A square polynomial matrix in \mathbb{R}^{n\times n}[x] is unimodular if its determinant is a constant and its inverse is also a polynomial matrix. A real matrix with an invertible determinant must have an inverse because the determinant being invertible means that it is non-zero, which is a necessary condition for a matrix to have an inverse.
  • #1
nahuel_pelado
2
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
 
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  • #2
Hint : What is a unimodular matrix?

Another hint: A square polynomial matrix U in [itex]\mathbb{R}^{n\times n}[x][/itex] is unimodular if the determinant [itex]\det U \neq 0[/itex] is a constant. The inverse of a unimodular polynomial matrix is again a polynomial matrix.
 
  • #3
Why does a real matrix with invertible determinant have to have an inverse?
 

1. What is modular arithmetic in matrices?

Modular arithmetic in matrices is the process of performing mathematical operations such as addition, subtraction, and multiplication on matrices with elements that are integers modulo a given number. This means that the operations are computed using the remainder when the elements are divided by the given number.

2. Why is modular arithmetic used in matrices?

Modular arithmetic in matrices is used to simplify calculations and make them more efficient. Additionally, it allows for the manipulation of large numbers without dealing with their exact values, which can be computationally expensive.

3. How is modular arithmetic applied in matrices?

Modular arithmetic is applied in matrices by performing the desired operation on each element of the matrix and then finding the remainder when divided by the given number. This can be done using a calculator or by hand.

4. What are the benefits of using modular arithmetic in matrices?

Modular arithmetic in matrices has several benefits, including making calculations more efficient and allowing for the manipulation of large numbers. It also has applications in cryptography, coding theory, and computer graphics.

5. Are there any limitations to using modular arithmetic in matrices?

One limitation of using modular arithmetic in matrices is that it may not always accurately represent the original values of the elements. This is because the remainder is used instead of the actual value. Additionally, division is not always possible in modular arithmetic, so not all operations can be performed on matrices using this method.

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