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Matrix Elementary Row Operations

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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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