Reduced row echelon form of a square matrix

In summary: BiPWhat about pivot rows and columns and zero rows in that case?There is no clear answer to that question. It depends on the specific case.
  • #1
Bipolarity
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I am wondering about the relation betwen RRE forms and identity matrices. Consider the reduced row echelon form of any square matrix. Must this reduced row echelon form of the matrix necessarily be an identity matrix?

I would suppose yes, but can this fact be proven? Could anyone provide an outline of the proof, or provide the link? Thanks much.

BiP
 
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  • #2
Bipolarity said:
I am wondering about the relation betwen RRE forms and identity matrices. Consider the reduced row echelon form of any square matrix. Must this reduced row echelon form of the matrix necessarily be an identity matrix?
Of course not. As a trivial example, take a square zero matrix, i.e. a square matrix such that all its elements are zeros. Or, more generally, any square marix with at least one zero row, or column. In fact, you can easily write down lots of square RRE matrices which are not identity matrices.

In general, a square matrix A is row equivalent to (i.e. its RRE is) the identity matrix of he same size if and only if A is invertible.
 
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  • #3
Erland said:
Of course not. As a trivial example, take a square zero matrix, i.e. a square matrix such that all its elements are zeros. Or, more generally, any square marix with at least one zero row, or column. In fact, you can easily write down lots of square RRE matrices which are not identity matrices.

In general, a square matrix A is row equivalent to (i.e. its RRE is) the identity matrix of he same size if and only if A is invertible.

What if I add the condition that the matrix square has no zero rows? Then is it necessarily the case that its RRE form is equivalent to the identity matrix (of the same size)?

BiP
 
  • #4
Bipolarity said:
What if I add the condition that the matrix square has no zero rows? Then is it necessarily the case that its RRE form is equivalent to the identity matrix (of the same size)?
No, for example
$$\begin{pmatrix}1 & 1 \\ 1 & 1\end{pmatrix}$$
 
  • #5
AlephZero said:
No, for example
$$\begin{pmatrix}1 & 1 \\ 1 & 1\end{pmatrix}$$

But how is that matrix in RRE form? The leading 1 in the second row is not strictly to the right of the leading 1 of the first row?

BiP
 
  • #6
Of course it's not in RRE form!

You asked if a square matrix with no zero rows always has an identity matrix for its RRE. That matrix has no zero rows. Reduce that matrix to RRE form and see what you get.

If you do that yourself, you might see WHY your idea is wrong (and even discover the right idea), which is more useful than just being told "your idea is wrong".
 
  • #7
AlephZero is saying to start with that matrix and then do row operations to put it into RRE form. You will find that you end up with a matrix that is not the identity matrix. Since the given matrix has no zero rows, it is a counter example to your modified question.
 
  • #8
I see! Thanks! The reduction gave me $$\begin{pmatrix}1 & 1 \\ 0 & 0\end{pmatrix}$$

What about if the RRE form of the matrix is a square matrix with no zero rows? In that case is the RRE form become an identity matrix?

BiP
 
  • #9
Bipolarity said:
What about if the RRE form of the matrix is a square matrix with no zero rows? In that case is the RRE form become an identity matrix?
Yes, that's right. It is easily verified if we carefully examine the definition of RRE and its consequences in the case of a square matrix. What about pivot rows and columns and zero rows in that case?
 

FAQ: Reduced row echelon form of a square matrix

1. What is reduced row echelon form (RREF) of a square matrix?

The reduced row echelon form (RREF) of a square matrix is a specific form of a matrix that has been reduced using row operations to make it easier to solve and interpret. In RREF, all leading coefficients (the first non-zero element in each row) are equal to 1, and all other elements in the same column are equal to 0.

2. How is reduced row echelon form different from row echelon form?

The main difference between reduced row echelon form and row echelon form is that in RREF, all elements below the leading coefficients are also equal to 0. In row echelon form, the elements below the leading coefficients can be non-zero. Additionally, in RREF, the leading coefficients must be equal to 1, while in row echelon form, they can be any non-zero number.

3. Why is reduced row echelon form useful?

RREF is useful because it simplifies a matrix and makes it easier to solve and interpret. It also provides a unique solution to a system of linear equations, if one exists. RREF can also be used to determine the rank of a matrix, which is useful in many applications, such as data analysis and machine learning.

4. How is reduced row echelon form calculated?

Reduced row echelon form is calculated using Gaussian elimination, which involves performing row operations on a matrix to transform it into the desired form. The specific steps for calculating RREF include finding the leading coefficients, making all elements below the leading coefficients equal to 0, and then using back substitution to solve for the remaining variables.

5. Can any square matrix be reduced to reduced row echelon form?

Not all square matrices can be reduced to reduced row echelon form. A matrix can only be reduced to RREF if it is invertible (has a non-zero determinant) and has a full rank (all rows are linearly independent). If a matrix does not meet these criteria, it may still be able to be reduced to row echelon form, but not RREF.

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