Possible Row Reduced Echelon Forms

In summary, the conversation discusses the difficulty in finding the possible combinations of reduced row echelon forms of nxn or mxn matrices. The professor suggests using exhaustion on every row as a method, but there may be simpler ways to figure it out. Additionally, the form of the matrix will depend on the number of free variables and leading variables after row reduction.
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This isn't homework.

I asked my professor for help on figuring out a way to know the possible combinations of reduced row echelon forms of nxn matrices, or mxn matrices.

He only could show me why it was really hard to find this out, not how to actually do it. His method was to use exhaustion on every row (i.e. consider every case on every row).

Are there simpler ways to figure this out?

Thanks for any help!


-F
 
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  • #2
I'm not sure I understand your question, but if your nxn-matrix is invertible, then its
reduced-row-echelon is row-equivalent to the identity. Otherwise, its form will have
to see with the number of free variables vs. leading variables left after row reduction.
 

What is a row reduced echelon form?

A row reduced echelon form (RREF) is a matrix in which the following conditions are met:

  • All rows consisting of only zeros are at the bottom of the matrix.
  • The first non-zero element in each row (called a leading entry) is always to the right of the leading entry in the row above it.
  • All leading entries are equal to 1.
  • All elements above and below a leading entry are equal to 0.

Why is it important to put a matrix in RREF?

Putting a matrix in RREF makes it easier to solve systems of linear equations and to find the rank and nullity of the matrix. It also helps in finding the inverse of a matrix and in determining if a set of vectors is linearly independent or dependent.

What are the steps for row reducing a matrix?

The steps for row reducing a matrix to RREF are:

  • Start with the leftmost column and find the first non-zero entry in the column.
  • Divide the entire row by the first non-zero entry to make it equal to 1.
  • Use row operations (such as multiplication and addition) to make all other entries in the column equal to 0.
  • Move on to the next column and repeat the process until the entire matrix is in RREF.

Can a matrix have more than one RREF?

Yes, a matrix can have more than one RREF. However, all RREFs of a given matrix will have the same number of leading entries, the same number of rows of all zeros at the bottom, and the same number of rows of leading zeros between leading entries. They will also have the same rank and nullity.

Is there a unique RREF for every matrix?

Yes, every matrix has a unique RREF. This is because the leading entries in the RREF are determined by the pivot positions in the augmented matrix, which are unique for every matrix. In addition, the leading entries determine the rest of the values in the RREF, making it unique for each matrix.

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