Inverse matrix with whole numbers

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Discussion Overview

The discussion revolves around the search for a 4x4 matrix composed solely of whole numbers that also has an inverse matrix with whole numbers. Participants explore potential algorithms and methods for constructing such matrices, as well as the properties that these matrices must satisfy.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant inquires about the existence of an algorithm to find n*n matrices with integer entries and integer inverses.
  • Another suggests that both matrices might only consist of zeros and ones.
  • A different participant mentions that there are matrices without zeros, as indicated by their teacher.
  • One participant recommends starting with simpler cases, such as 2x2 matrices, to explore the properties before extending to larger matrices.
  • It is noted that the inverse of a matrix will have integer entries if its determinant is 1, provided that any common factor has been factored out.
  • Another participant challenges the idea that matrices will only contain zeros and ones by providing a counterexample with different integer entries.
  • There is a question about whether the property of having integer inverses holds true for matrices with determinants of ±1.
  • Several participants express difficulty in finding a specific 4x4 matrix that meets the criteria and request examples.
  • One suggestion is to start with the 4x4 identity matrix and apply transformations that preserve the determinant's value.

Areas of Agreement / Disagreement

Participants express differing views on the types of matrices that can satisfy the conditions, with some suggesting limitations to zeros and ones, while others argue for the possibility of matrices containing other integers. The discussion remains unresolved regarding the specific construction of a 4x4 matrix with the desired properties.

Contextual Notes

Participants have not reached a consensus on the methods or examples for constructing the desired matrices, and there are various assumptions about the properties of the matrices being discussed.

posuchmex
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Hello, how to find matrix 4x4 which only contains whole numbers and has inverse matrix with whole numbers only aswell?

Is there algorithm to find such matrix of n*n?

Thanks.
 
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any ideas please?
 
Both matrices will probably contain only zeros and ones.
 
our teacher said there are some without zeros
 
Well I would use from Polya's 'How to solve it' the recommendation of instead of staring with no ideas at the problem in its generality, start with a simple example.

Use only 1's and 0's for simplicity. Anyway I suspect that any other example will be a multiple or something simply related to such a matrix.

Instead of thinking about 4X4 matrices yet, attack a simpler case - 2X2 matrices. You can surely find several 2X2 matrices that have your property.

Then does that suggest a plan for extending to construction of suitable 3X3 matrices? If you can do that you will probably be able to do it for 4X4 too.
 
The inverse of a matrix can be written by replacing each entry by its "cofactor" (the determinant of the matrix you get by dropping the entire row and column of the entry) divided by the determinant of the matrix. Assuming that you have already factored out any factor common to all entries in the matrix, the inverse of a matrix with integer entries will have integer entries if and only if its determinant is 1.
 
SteamKing said:
Both matrices will probably contain only zeros and ones.

not so:

[1 1][2 -1]...[1 0]
[1 2][-1 1] = [0 1]
 
Halls of Ivy: isn't this true for matrices with determinant ± 1?
 
Last edited:
i can't find 4x4 this matrix

can you show me one please
 
Last edited:
  • #10
posuchmex said:
i can't find 4x4 this matrix

can you show me one please

Can you find a 4x4 matrix with integer entries whose determinant is 1??
 
  • #11
"i can't find 4x4 this matrix"

Where did you leave it last? Retrace your steps, and you may find it... or:


Start with the 4x4 identity, and apply transformations to the rows that preserve the value of
the determinant.
 

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