Listing Possible RREF Matrix Combinations

In summary, the conversation is discussing the possible combinations that can result from performing reduced row echelon form on a 2 x 3 matrix using the values 1, 0, and * (representing any number). The speaker has found 13 different combinations, but is unsure if this is the correct amount or if any are missing. They are seeking help in determining the correct formula for finding all possible rref matrices for an m x n matrix. Another person suggests starting by listing all the constraints that apply to such a matrix and gives a formula of 7 for the 2 x 3 case.
  • #1
MDS
1
0

Homework Statement


Using the values 1, 0, and *(to represent any number), list all the possible combinations that could result from performing reduced row echelon form on a 2 x 3 matrix.




Homework Equations


Instructor 2 x 2 matrix example:
2ci7nky.png


The Attempt at a Solution


I have found 13 different combinations:
wsa2qp.png


but we have no examples beyond the 2 x 2 matrix so I haven't been able to attempt to model an equation(since I can't find a theorem or definition that models this) to predict the number of combinations I should find for a 2 x 3 matrix.

Any help in determining if this is the correct amount of combinations for a 2 x 3 matrix that has been simplified to RREF or what ones I may be missing would be appreciated.
 
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  • #2
how to find all possible rref matrix for m*n matrix. how I create a formula for m*n matrix
 
  • #3
Aryan patel said:
how to find all possible rref matrix for m*n matrix. how I create a formula for m*n matrix
Start by listing all the constraints that apply to such a matrix.

By the way, I believe @MDS was overcounting in the 2x3 case. You should not count separately a case which can be got by replacing some * with 0s and 1s from another case.
E.g
1**
000
subsumes
100
000

For 2x3, the general formula I get gives 7.
 
Last edited:
  • #4
  • #5
Dhruv shrivastava said:
@haruspex what's your general formula
Do you also have this question as homework? If so, you need to show some attempt, per forum rules.
 

1. What is a RREF matrix combination?

A RREF matrix combination is a matrix that has been reduced to row-echelon form (REF) and then further reduced to reduced row-echelon form (RREF). This is the most simplified form of a matrix, making it easier to perform calculations and solve equations.

2. How many RREF matrix combinations are possible?

The number of possible RREF matrix combinations depends on the size of the original matrix. For an n x n matrix, there are 2^n possible combinations. This means that as the matrix size increases, the number of possible combinations also increases exponentially.

3. What is the process for finding RREF matrix combinations?

To find the RREF matrix combination of a given matrix, you must perform row operations to reduce the matrix to REF, and then continue to reduce it to RREF. The row operations include swapping rows, multiplying a row by a nonzero constant, and adding a multiple of one row to another.

4. Why is it important to find RREF matrix combinations?

RREF matrix combinations are important because they represent the most simplified form of a matrix. This makes it easier to perform calculations and solve equations involving the matrix. Additionally, RREF matrices are useful in various applications such as computer graphics, engineering, and statistics.

5. Are there any limitations to finding RREF matrix combinations?

Yes, there are limitations to finding RREF matrix combinations. The most significant limitation is that not all matrices can be reduced to RREF. This is because some matrices are singular or have linearly dependent rows, which cannot be reduced using row operations. Another limitation is that finding RREF matrix combinations can be a time-consuming process, especially for larger matrices.

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