**Up:** *Math 15 Selected Class Notes*

# Gauss-Jordon Elimination Notes:

The procedure for *Guass-Jordan Elimination* is as follows:
- Find the leftmost column that is not all zeros.

- Interchange the top row (if necessary) with another row to bring a
non-zero entry to the top of the column.

- If the top entry is
*a*, then multiply the top row by 1/*a*
to form a leading 1 in that row.

- Add multiples of this row to the other rows so that all other rows have
a 0 in this column.

- Cover up the top row and go back to step 1, considering only the rows
bewlo this one (until step 4). Continue until the matrix is in reduced
row-echelon form.

## Example:

Solve for *x*, *y*, and *z* in
2*y* - 3*z* = 2
2*x* + *z* = 3
*x* - *y* + 3*z* = 1.
+- -+
| 0 2 -3 | 2 | write the system as an augmented matrix
| 2 0 1 | 3 |
| 1 -1 3 | 1 |
+- -+
+- -+
| 1 -1 3 | 1 | interchange first and third row
| 2 0 1 | 3 | (to make top left entry non-zero)
| 0 2 -3 | 2 |
+- -+
+- -+
| 1 -1 3 | 1 | Add -2 times first row to second row
| 0 2 -5 | -1 | (to get 0 in first column of row 2)
| 0 2 -3 | 2 |
+- -+
+- -+
| 1 -1 3 | 1 | Divide second row by 2
| 0 1 -5/2 | -1/2 | (to get a leading 1 in row 2)
| 0 2 -3 | 2 |
+- -+
+- -+
| 1 0 1/2 | 3/2 | Add second row to first
| 0 1 -5/2 | -1/2 | Add -2 times second row to third
| 0 0 2 | 1 | (to get 0's in the second column)
+- -+
+- -+
| 1 0 1/2 | 3/2 | Divide third row by 2
| 0 1 -5/2 | -1/2 | (to get a leading 1)
| 0 0 1 | 1/2 |
+- -+
+- -+
| 1 0 0 | 5/4 | Add -1/2 times third row to first row
| 0 1 0 | 7/4 | Add 5/2 times third row to second row
| 0 0 1 | 1/2 | (to get 0's in third column)
+- -+

This is now in reduced row-echelon form, so we can read off the answer:
*x* = 5/4, *y* = 7/4, and
*z* = 1/2.
Check that the answer satisfies the initial equations (in case we made
arithmatic errors):

2*y* - 3*z* = 2(7/4) - 3(1/2) = 7/2 - 3/2 = 4/2 = 2

2*x* + *z* = 2(5/4) + 1/2 = 5/2 + 1/2 = 6/2 = 3

*x* - *y* + 3*z* = (5/4) - (7/4) + (1/2) = -(2/4) + 3/2
= -(1/2) + 3/2 = 2/2 = 1

All of these check out, so our solution is correct.

**Up:** *Math 15 Selected Class Notes*

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Created: Feb 6 1998 ---
Last modified: Nov 11, 1998 10:43:26 AM