Elimination using Matrices

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Section 8.6

• Elimination using Matrices

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Matrix Method

• The method computers use.
• The equations need to be in standard form.
• The coefficients and constants are translated
  into a rectangle array.
• Make the rectangle array into row echelon form.
• Find answer by using back substitution.

3
Standard Form

• Write the linear equation in the form
• Ax By C
• If variables are different, go in alphabet order.
• A, B and C do not have any restrictions, but life
  is easier if they are integers .., -2, -1, 0, 1,
  2, ..

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Standard Form to Rectangle Array

• Write the standard form of system of linear
  equations
• In rectangle array form

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Example

• Write the system of linear equations in standard
  form and in rectangle array
• Standard form
• Rectangle array

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Row Echelon Form

• The values on the diagonal need to be ones.
• The values below the diagonal need to be zeros.
• The other values can be any number, using lower
  class letters because they could change values.

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Back Substitution

• Given the row echelon form.
• Rewrite it in standard form.
• Solve the bottom equation then the top equation
• y f
• x by c x bf c x c - bf

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Example

• Given the matrix find the values for x and y
• 1. Write the bottom row as an equation
• 0x 0y -1
• 2. Solve y -1
• 3. Write the top row as an equation
• 1x 3y 0
• 4. Substitute the answer we found for y
• 1X 3(-1) 0
• 5. Solve X 3
• 6. Write answer in ordered form (3, -1)

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Rules to make the Row Echelon Form

• The following operations produces a row
  equivalent matrix
• 1. Interchanging any two rows.
• 2. Multiplying all elements of a row by a nonzero
  constant.
• 3. Adding two rows together.
• You can blend rules together, especially 2 and 3
• Each step needs to include the proper rule.

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Interchanging two rows

• The proper format for this rule is
• i and j are the specific rows you will swap
• Example. Write the matrix in row echelon
  form.

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Multiplying a row by a constant

• The proper format for this rule is
• i is the row
• c is the constant
• The constant will be multiplied to all values in
  the row.
• Example Write the matrix in row echelon form.

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Adding two rows together

• The proper format for this rule is
• i and j are the two rows to be added
• i is the row you will be placing the answer into
• Example Write the matrix in row echelon form

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Example

• Solve the system of linear equations.

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Example

• Solve the system of linear equations

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No Solution and Infinite Solutions

• If the matrix looks like then
  you have a no solution.
• If the matrix looks like then you
  have an infinite solutions.

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Example

• Solve the system of linear equations

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Homework

• Section 8.7 7, 8, 9, 10, 11, 12