Section 16.3: Matrix Solution of a System of Equations

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16.3 Outline

  1. Definition of a matrix
    1. subscripts
    2. double subscripts
    3. array
    4. matrix
    5. rows and columns
    6. square matrix
    7. order/dimension
  2. Matrix form of a system of equations
    1. augmented matrix
    2. diagonal form
  3. Elementary row operations and pivoting
    1. elementary row operations
    2. equivalent matrices
    3. elementary row operation 1; RowSwap
    4. elementary row operation 2; Row+
    5. elementary row operation 3; *Row
    6. elementary row operation 4; *Row+
    7. pivoting
  4. Gauss-Jordan elimination
    1. row-reduced form
    2. procedure

 

16.3 Essential Ideas

There are four elementary row operations for producing equivalent matrices:

  1. RowSwap  Interchange any two rows.
  2.  Row+ (Row addition); add a row to any other row.
  3. *Row (Scalar multiplication); multiply (or divide) all the elements of a row by the same nonzero real number.
  4. *Row+ Multiply all the entries of a row (pivot row) by a nonzero real number and add each resulting product to the corresponding entry of another specified row (target row).

These elementary row operations are used together in a process called pivoting: which means

  1. Divide all entries in the row in which the pivot appears (called the pivot row) by the nonzero pivot element so that the pivot entry becomes a 1. This uses elementary row operation 3.
  2. Obtain zeros above and below the pivot element by using elementary row operation 4.

GAUSS-JORDAN ELIMINATION

Step 1: Select as the first pivot the element in the first row, first column, and pivot.
Step 2: The next pivot is the element in the second row, second column; pivot.
Step 3: Repeat the process until you arrive at the last row, or until the pivot element is a& zero. If it is a zero and you can interchange that row with a row below it, so that the pivot element is no longer a zero, do so and continue. If it is zero and you cannot interchange rows so that it is not a zero, continue with the next row. The final matrix is called the row-reduced form.