Elementary Row Operations
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.
- Interchanging two rows of the matrix. Notation
means to interchange row 
and row
. - Multiplying a row of the matrix by a scalar. Notation
means to multiply row 
by
, 
. - Add a scalar multiple of a row to another row. Notation
means to add 
times the row
to row
,
.
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 Matrix Type 2: Multiplying a row of the matrix by a scalar
,
, 
Elementary Matrix Type 3: Add a scalar multiple of a row to another row
,
, 
,
,
, 
Example:
Suppose we have matrix
Applying elementary matrix type 3 of 
produces row equivalent matrix
Applying elementary matrix type 2 of 
to the last result produces row equivalent matrix
Applying elementary matrix type 1 of 
to the last result produces row equivalent matrix
See also: Matrix RREF, Matrix Inverse using Gauss Jordan, matrix rank
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Teknomo, Kardi (2011) Linear Algebra tutorial. https:\\people.revoledu.com\kardi\tutorial\LinearAlgebra\