Gauss-Jordan Matrix Elimination

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Gauss-Jordan Matrix Elimination

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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

• a1 x b1 y c1 z d1
• a2 x b2 y c2 z d2
• a3 x b3 y c3 z d3

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 Matrix
2. Switch row 1 with row 2
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3. Multiply Row 1 by -3 and add Row 2
-3 3 -6 -12 3 2 -1 3 0 5 -7 -9
R1(-3)
R2
R2
R1(-3) R2 ? R2
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4. Multiply Row 1 by -2 and add Row 3
-2 2 -4 -8 2 3 -1 3 0 5 -5 -5
R1(-2)
R3
R3
R1(-2) R3 ? R3
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5. Switch Row 2 with Row 3
R2 R3
6. Multiply Row 2 by 1/5
R2 (1/5 ) ? R2
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7. Add Row 2 to Row 1
R1 R2 ? R1
8. Multiply Row 2 by -5 and Add Row 3
R2 (-5) R3 ? R3
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9. Multiply Row 3 by -1/2
R3 ( -1/2 ) ? R3
10. Add Row 3 and Row 2
R3 R2 ? R2
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11. Multiply Row 3 by -1 and add Row 1
R3(-1) R1 ? R1
Final Answer
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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