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Gauss-Jordan Matrix Elimination Brought to you by Tutorial Services The Math Center

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Gauss-Jordan Matrix Elimination A method that can be used to solve systems of linear equations involving two or more variables. To do so, the system must be changed first, to an augmented matrix.

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Augmented Matrix a 1 x +b 1 y +c 1 z =d 1 a 2 x +b 2 y +c 2 z =d 2 a 3 x +b 3 y +c 3 z =d 3 System of Equations Augmented Matrix

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Example System of Equations Augmented Matrix

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Elementary Row Operations 1.Interchanging two rows. 2.Adding one row to another row, or multiplying one row by a constant first and then adding it to another. 3.Multiplying a row by any constant different from zero.

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Gauss-Jordan Matrix Elimination Goal In order to solve the system of equations, a series of steps needs to be followed using the elementary row operations. The reduced matrix should end up being the identity matrix.

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Identity Matrix Identity Matrix for a 3 x 3 Identity Matrix for a 4 x 4

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Solving the System 1. Write as an augmented Matrix2. Switch row 1 with row 2

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3. Multiply Row 1 by -3 and add Row 2 R 1 (-3) -33-6-12 + 32-1 3 05-7 -9 R2R2 R2R2 R 1 (-3) + R 2 R 2

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4. Multiply Row 1 by -2 and add Row 3 R 1 (-2) -22-4-8 + 23-1 3 05-5 -5 R3R3 R3R3 R 1 (-2) + R 3 R 3

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5. Switch Row 2 with Row 3 6. Multiply Row 2 by 1/5 R 2 (1/5 ) R 2 R 2 R 3

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7. Add Row 2 to Row 1 R 1 + R 2 R 1 8. Multiply Row 2 by -5 and Add Row 3 R 2 (-5) + R 3 R 3

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10. Add Row 3 and Row 2 9. Multiply Row 3 by -1/2 R 3 ( -1/2 ) R 3 R 3 + R 2 R 2

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Final Answer 11. Multiply Row 3 by -1 and add Row 1 R 3 (-1) + R 1 R 1

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Gauss Jordan Handouts and Links Gauss Jordan Method Handout Adding and Subtracting Matrices Workshop Adding and Subtracting Matrices Handout Multiplying Matrices Workshop Multiplying Matrices Handout Inverse Matrix Handout