Inverse matrix with whole numbers

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SUMMARY

This discussion focuses on finding a 4x4 matrix composed solely of whole numbers that also has an inverse matrix with whole numbers. Participants suggest starting with simpler cases, such as 2x2 matrices, to build understanding before tackling 4x4 matrices. The key insight is that for a matrix with integer entries to have an integer inverse, its determinant must equal 1. The conversation emphasizes the importance of transformations that preserve the determinant value when constructing such matrices.

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  • Understanding of matrix operations and properties
  • Familiarity with determinants and their significance
  • Knowledge of matrix inverses and cofactor calculations
  • Basic experience with integer matrices and their transformations
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  • Research methods for calculating determinants of 4x4 matrices
  • Explore algorithms for generating integer matrices with specific properties
  • Learn about matrix transformations that preserve determinants
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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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