## 16.3 Outline

- Definition of a matrix
- subscripts
- double subscripts
- array
- matrix
- rows and columns
- square matrix
- order/dimension

- Matrix form of a system of equations
- augmented matrix
- diagonal form

- Elementary row operations and pivoting
- elementary row operations
- equivalent matrices
- elementary row operation 1; RowSwap
- elementary row operation 2; Row+
- elementary row operation 3; *Row
- elementary row operation 4; *Row+
- pivoting

- Gauss-Jordan elimination
- row-reduced form
- procedure

## 16.3 Essential Ideas

There are four elementary row operations for producing equivalent matrices:

**RowSwap**Interchange any two rows.-
**Row+**(Row addition); add a row to any other row. ***Row**(Scalar multiplication); multiply (or divide) all the elements of a row by the same nonzero real number.***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

- 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.
- 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.