The Nature of Mathematics, 12th Edition
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Section 16.3: Matrix Solution of a System of Equations

16.3 Outline

A.  Definition of a matrix
     1. subscripts
     2. double subscripts
     3. array
     4. matrix
     5. rows and columns
     6. square matrix
     7. order/dimension
B. Matrix form of a system of equations
     1. augmented matrix
     2. diagonal form
C. 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
D.  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.