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Guilherme Mazui

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What is the Difference Between Echelon Form and Reduced Echelon Form?

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The difference between Echelon Form and Reduced Echelon Form lies in the values of their non-zero entries and the strictness of their requirements. Here are the main differences:

  1. Values of non-zero entries: In Echelon Form, the matrix can contain any value as its non-zero entries, while in Reduced Echelon Form, only the number 1 is allowed as the leading coefficients.
  2. Strictness of requirements: Echelon Form requires that the first non-zero value of each row after the first one can be any value as long as it's not 0. On the other hand, Reduced Echelon Form imposes additional conditions on the leading coefficients, making it a stricter version of Echelon Form.
  3. Uniqueness: Reduced Echelon Form is unique, meaning that row-reduction on a matrix will produce the same answer no matter how it is performed.

Both Echelon Form and Reduced Echelon Form are used to solve systems of linear equations, but the choice between using one or the other depends on the specific problem and the desired level of strictness.

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  • What is the Difference Between Echelon Form and Reduced Echelon Form?
  • Comparative Table: Echelon Form vs Reduced Echelon Form

    • Comparative Table: Echelon Form vs Reduced Echelon Form

    Here is a table comparing Echelon Form and Reduced Echelon Form:

    Echelon Form Reduced Echelon Form
    A matrix in echelon form meets the following requirements:
    - The first non-zero number in the row (called a leading coefficient) is 1.
    - The first non-zero number in each subsequent row is to the right of the first non-zero number in the previous row.
    - Rows consisting of all zeros are at the bottom of the matrix.    		 A matrix in reduced echelon form meets the following requirements:
    - The first non-zero number in the first row (the leading entry) is the number 1.
    - The second row also starts with the number 1, which is further to the right than the leading entry in the first row. For every subsequent row, the number 1 must be further to the right.
    - All other entries in the matrix are zeros.

    In summary, the reduced echelon form is a stricter version of the echelon form, requiring additional conditions for the leading coefficients and having all non-pivot entries equal to zero.

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    Guilherme Mazui

    Guilherme Mazui is graduated in journalism from the Federal University of Minas Gerais (UFMG) and a master's degree in Communication from the University of São Paulo (USP). In addition, he has experience in advertising writing and has worked as a content editor in several companies.