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Resources on Linear Algebra

Elementary Row Operations

By Kardi Teknomo, PhD.
LinearAlgebra

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In linear algebra, there are 3 elementary row operations. The same operations can also be used for column (simply by changing the word “row” into “column”). The elementary row operations can be applied to a rectangular matrix size m by n.

  1. Interchanging two rows of the matrix. Notation Elementary row operationsmeans to interchange row Elementary row operationsand rowElementary row operations.
  2. Multiplying a row of the matrix by a scalar. Notation Elementary row operationsmeans to multiply row Elementary row operationsbyElementary row operations, Elementary row operations.
  3. Add a scalar multiple of a row to another row. Notation Elementary row operationsmeans to add Elementary row operationstimes the rowElementary row operations to rowElementary row operations,Elementary row operations.

Applying the three elementary row operations to a matrix will produce row equivalent matrix. When we view a matrix as augmented matrix of a linear system, the three elementary row operations are equivalent to interchanging two equations, multiplying an equation by a non-zero constant and adding a scalar multiple of one equation to another equation. Two linear systems are equivalent if they produce the same set of solutions. Since a matrix can be seen as a linear system, applying the 3 elementary row operations does not change the solutions of that matrix.

The 3 elementary row operations can be put into 3 elementary matrices. Elementary matrix is a matrix formed by performing a single elementary row operation on an identity matrix. Multiplying the elementary matrix to a matrix will produce the row equivalent matrix based on the corresponding elementary row operation.

Elementary Matrix Type 1: Interchanging two rows of the matrix
Elementary row operations,Elementary row operations,Elementary row operations

Elementary Matrix Type 2: Multiplying a row of the matrix by a scalar
Elementary row operations,Elementary row operations , Elementary row operations

Elementary Matrix Type 3: Add a scalar multiple of a row to another row

Elementary row operations,Elementary row operations , Elementary row operations,
Elementary row operations,Elementary row operations , Elementary row operations

Example:
Suppose we have matrix
Elementary row operations
Applying elementary matrix type 3 of Elementary row operationsproduces row equivalent matrix
Elementary row operations
Applying elementary matrix type 2 of Elementary row operationsto the last result produces row equivalent matrix
Elementary row operations

Applying elementary matrix type 1 of Elementary row operationsto the last result produces row equivalent matrix
Elementary row operations

See also: Matrix RREF, Matrix Inverse using Gauss Jordan, matrix rank

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Preferable reference for this tutorial is

Teknomo, Kardi (2011) Linear Algebra tutorial. http:\\people.revoledu.com\kardi\ tutorial\LinearAlgebra\

 

 
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