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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