# Inverse matrix with whole numbers

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.

SteamKing
Staff Emeritus
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Both matrices will probably contain only zeros and ones.

our teacher said there are some without zeros

epenguin
Homework Helper
Gold Member
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.

HallsofIvy
Homework Helper
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.

Deveno
Both matrices will probably contain only zeros and ones.

not so:

[1 1][2 -1]....[1 0]
[1 2][-1 1] = [0 1]

Bacle2
Halls of Ivy: isn't this true for matrices with determinant ± 1?

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i cant find 4x4 this matrix

can you show me one please

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micromass
Staff Emeritus
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i cant find 4x4 this matrix

can you show me one please

Can you find a 4x4 matrix with integer entries whose determinant is 1??

Bacle2