What is the difference between row echelon and reduced row echelon?

Question:

What is the difference between row echelon and reduced row echelon?

Reduced Row Echelon Form


Using row operations on a matrix, we can obtain the reduced row echelon form of the given matrix, when we obtain unit pivots with zeros entries above and below each pivot.

Performing Gaussian elimination on a matrix, we will obtain the reduced row echelon form by starting with the first nonzero entry on the first row, doing row operations to obtain zeros under the first nonzero entry.

Next, we create the second pivot, by choosing the first nonzero on the second row (or interchange rows to obtain it, if it is not there) and create zero entries under the second pivot.

Repeat the process until we obtain an upper triangular matrix.

Next, we will create zeros above each pivot, to obtain the reduced row echelon form.

Note that a row echelon form is not unique, but the reduced row echelon form is unique.

Answer and Explanation:


The reduced row echelon form is a matrix in a row echelon form with each leading 1 being the only nonzero entry in its column.

To obtain a row echelon form of a matrix, we do row operations on the matrix until we obtain a matrix with

- all nonzero rows being above any rows of all zeros,

- each leading entry of a row is in a column to the right of the leading entry of the row above it

- all entries of a column below a leading entry are zeros.

So, {eq}\displaystyle \boxed{\text{ the reduced row echelon form is a row echelon form with unit pivots and zero entries above the pivots}} {/eq}

For example, the form {eq}\displaystyle \left[ \begin{array}{ccccc} \boxed{a_{11}}&*&*&*&*&* \\ 0&\boxed{a_{22}}&*&*&*&*\\ 0&0&0&\boxed{a_{33}}&*&*\\ 0&0&0&0&\boxed{a_{44}} &* \end{array} \right] {/eq} is in row echelon form and

the reduced row echelon form is {eq}\displaystyle \left[ \begin{array}{cccccc} \boxed{1}&0&*&0&0&* \\ 0&\boxed{1}&*&0&0&*\\ 0&0&0&\boxed{1}&0&*\\ 0&0&0&0&\boxed{1} &* \end{array} \right]. {/eq}

For example, the reduced row echelon and a row echelon forms of the matrix {eq}\displaystyle \left[ \begin{array}{ccccc} 2&4&0&2 \\ 1&2&1&0 \\ 1&1&0&0 \end{array} \right] {/eq} are obtained as below.

{eq}\displaystyle \begin{align} &\left[ \begin{array}{ccccc} 2&4&0&2 \\ 1&2&1&0 \\ 1&1&0&0 \end{array} \right] \overset{-\frac{1}{2}R_1+R_2}{\implies }\left[ \begin{array}{cccc} 2&4&0&2 \\ 0&0&1&-1 \\ 1&1&0&0 \end{array} \right] \\\\ &\overset{-\frac{1}{2}R_1+R_3}{\implies }\left[ \begin{array}{cccc|c} 2&4&0&2 \\ 0&0&1&-1 \\ 0&-1&0&-1 \end{array} \right] \overset{R_2 \text{ interchanged with }R_3}{\implies }\left[ \begin{array}{cccc|c} \boxed{2}&4&0&2 \\ 0&\boxed{-1}&0&-1\\ 0&0&\boxed{1}&-1 \end{array} \right] \text{ which is a row echelon form }\\\\ &\overset{\frac{1}{2}R_1}{\implies }\left[ \begin{array}{cccc|c} 1&2&0&1 \\ 0&-1&0&-1\\ 0&0&1&-1 \end{array} \right] \overset{-R_2}{\implies }\left[ \begin{array}{cccc|c} 1&2&0&1 \\ 0&1&0&1\\ 0&0&1&-1 \end{array} \right]\overset{-2R_2+R_1}{\implies }\left[ \begin{array}{cccc|c} \boxed{1}&0&0&-1 \\ 0&\boxed{1}&0&1\\ 0&0&\boxed{1}&-1 \end{array} \right], \text{ which is the reduced row echelon form matrix}. \end{align} {/eq}


Learn more about this topic:

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Inconsistent and Dependent Systems: Using Gaussian Elimination

from Algebra II Textbook

Chapter 10 / Lesson 8
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