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Reduced Row-Echelon Form: Definition & Examples

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  • 0:03 Matrix Review
  • 1:10 Row-Echelon Form
  • 1:38 Reduced Row-Echelon Form
  • 2:55 A Note on Calculators
  • 3:17 Lesson Summary
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Lesson Transcript
Instructor: Jasmine Cetrone

Jasmine has taught college Mathematics and Meteorology and has a master's degree in applied mathematics and atmospheric sciences.

In this lesson, we'll look at one of the most useful forms of a matrix: the reduced row-echelon form. We'll review the definition of reduced row-echelon form and look at several examples.

Matrix Review

Before we get into the reduced row-echelon form of a matrix, we should make sure we're all up to speed on the basics of matrices. First, why do we care about matrices? Matrices can be used to solve a variety of applied math and statistics problems, including solving a system of linear equations, processing computer graphics, even encrypting messages so that information stays secret!

So, what is a matrix? A matrix is an array (or table) of numbers. Think of it like a rectangular grid where the numbers are neatly arranged in rows and columns. We talk about the size of a matrix by how many rows and columns it has (in that order). A 4 by 5 matrix (also written 4 x 5) has 4 rows and 5 columns, like the matrix shown.

a 4x5 matrix

When we classify a matrix in different forms, we're talking about the arrangement of the numbers within the matrix. Certain patterns can be advantageous for solving specific problems. For solving a system of linear equations, the reduced row-echelon form of a matrix is what we want.

Row-Echelon Form

Before we discuss reduced row-echelon form, we should talk briefly about regular row-echelon form. A matrix is in row-echelon form if all of the following conditions are true:

  • Each row contains only zeros until the first non-zero entry
  • Each leading non-zero entry of a row is in a column to the right of all the leading non-zero entries in the rows above it
  • Any rows that contain all zeros are below any rows that have non-zero entries

Reduced Row-Echelon Form

Reduced row-echelon form is more restrictive in its construction. In addition to being in row-echelon form, we have two extra criteria: the leading entries have to be the number 1, and above and below these leading 1 entries we can only have zero entries.

Putting this all together, we can say that a matrix is in reduced row-echelon form if all of the following conditions are true:

  • Each row contains only zeros until the first non-zero entry, which must be a 1
  • Each leading 1 of a row is in a column to the right of all the leading 1 entries in the rows above it
  • Any rows that contain all zeros are below any rows that have a leading 1
  • All entries above and below the leading 1 must be zero

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