# Elementary Row Operations to Perform on Matrices

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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