Inverse matrix with whole numbers

In summary: If the determinant doesn't change, you've found the inverse!In summary, the inverse of a matrix with integer entries can be found if and only if its determinant is 1.
  • #1
posuchmex
5
0
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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  • #2
any ideas please?
 
  • #3
Both matrices will probably contain only zeros and ones.
 
  • #4
our teacher said there are some without zeros
 
  • #5
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.
 
  • #6
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.
 
  • #7
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]
 
  • #8
Halls of Ivy: isn't this true for matrices with determinant ± 1?
 
Last edited:
  • #9
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.
 

1. What is an inverse matrix with whole numbers?

An inverse matrix with whole numbers is a square matrix that, when multiplied by the original matrix, results in the identity matrix. It is represented by A-1 and is used to solve systems of equations and perform other mathematical operations.

2. How do you find the inverse of a matrix with whole numbers?

To find the inverse of a matrix with whole numbers, you can use the Gauss-Jordan elimination method. This involves performing row operations on the matrix until it is in reduced row-echelon form, with the identity matrix on the left side. The resulting matrix is the inverse of the original matrix.

3. Can an inverse matrix with whole numbers exist for any square matrix?

No, an inverse matrix with whole numbers only exists for square matrices that are invertible. This means that the matrix has a non-zero determinant and all its rows and columns are linearly independent.

4. What is the purpose of finding the inverse of a matrix with whole numbers?

The inverse of a matrix with whole numbers has many practical applications, such as solving systems of linear equations, finding the solution to a system of differential equations, and performing transformations in linear algebra. It also allows for the efficient computation of the original matrix's determinant and other matrix operations.

5. Is finding the inverse of a matrix with whole numbers the same as finding the reciprocal of a number?

No, finding the inverse of a matrix with whole numbers is not the same as finding the reciprocal of a number. Inverse matrices involve a more complex process of finding the matrix that, when multiplied by the original matrix, gives the identity matrix. Reciprocals, on the other hand, are simply the multiplicative inverse of a single number.

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