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# Matrix Elementary Row Operations

Elementary row operations are performed on the augmented matrix of a system of linear equations. These operations produce a new augmented matrix corresponding to a new equivalent system of linear equations. Algorithms that use matrix elementary row operations to solve systems of linear equations are Gaussian Elimination with Back-Substitution and Gauss-Jordan Elimination.

There are three elementary row operations:

1) The interchanging of two rows,

2) The multiplying a row by a non-zero constant, and

3) The adding of a multiple of a row to another row.

Interchanging of Two Rows:

 3×4 Column 1 Column 2 Column 3 Column 4 Row 1: 3 4 5 6 Row 2: −2 1 0 3 Row 3: 4 2 2 4

Interchanged Rows 1 and 2:

 3×4 Column 1 Column 2 Column 3 Column 4 Row 1: −2 1 0 3 Row 2: 3 4 5 6 Row 3: 4 2 2 4

Multiplying a Row by a Non-zero Constant:

 3×4 Column 1 Column 2 Column 3 Column 4 Row 1: −4 0 2 6 Row 2: 2 −4 4 1 Row 3: −4 0 2 6

Multiply Row 2 by 1/2:

 3×4 Column 1 Column 2 Column 3 Column 4 Row 1: −4 0 2 6 Row 2: 1 −2 2 1/2 Row 3: −4 0 2 6

Adding a Multiple of a Row to Another Row:

 3×4 Column 1 Column 2 Column 3 Column 4 Row 1: 1 2 −6 3 Row 2: 0 3 −2 −1 Row 3: 5 2 1 −2

Add -2 times the First Row to Third Row:

 3×4 Column 1 Column 2 Column 3 Column 4 Row 1: 1 2 −6 3 Row 2: 0 3 −2 −1 Row 3: 3 −2 13 −8
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