Math 15 (Notes)

# Gauss-Jordon Elimination Notes:

The procedure for Guass-Jordan Elimination is as follows:
1. Find the leftmost column that is not all zeros and swap its row with the top row.
2. Make the leading entry in the top row a "1". (If the top entry is a, then multiply the top row by 1/a).
3. Use the top row to make all the other entries in the column containing the leading one into zeros.
4. Ignoring the top row, repeat steps 1 to 4 until there are no more leading ones.

## Example:

Solve for x, y, and z in:
\begin{eqnarray} 2y - 3z &= 2\cr 2x + z &= 3\cr x - y + 3z &= 1\cr \end{eqnarray}
Solution:
 \Matrix{ 0& 2& -3& & 2\cr 2& 0& 1& \|& 3\cr 1& -1& 3& & 1\cr } write the system as an augmented matrix \Matrix{ 1& -1& 3& & 1\cr 2& 0& 1& \|& 3\cr 0& 2& -3& & 2\cr } interchange first and third rows (to make top left entry non-zero) \Matrix{ 1& -1& 3& & 1\cr 0& 2& -5& \|& 1\cr 0& 2& -3& & 2\cr } Add -2 times first row to second row (to get 0 in first column of row 2) \Matrix{ 1& -1& 3& & 1\cr 0& 1& -5/2& \|& 1/2\cr 0& 2& -3& & 2\cr } Divide second row by 2 (to get a leading 1 in row 2) \Matrix{ 1& 0& 1/2& & 3/2\cr 0& 1& -5/2& \|& 1/2\cr 0& 0& 2& & 1\cr } Add second row to first row Add -2 times second row to third (to get 0's in the second column) \Matrix{ 1& 0& 1/2& & 3/2\cr 0& 1& -5/2& \|& 1/2\cr 0& 0& 1& & 1/2\cr } Divide third row by 2 (to get a leading 1) \Matrix{ 1& 0& 0& & 5/4\cr 0& 1& 0& \|& 7/4\cr 0& 0& 1& & 1/2\cr } Add -1/2 times third row to first row Add 5/2 times third row to second row (to get 0's in third column)
Now 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):

\begin{array}{l} 2y - 3z = 2(7/4) - 3(1/2) = 7/2 - 3/2 = 4/2 = 2\cr 2x + z = 2(5/4) + 1/2 = 5/2 + 1/2 = 6/2 = 3\cr x - y + 3z = (5/4) - (7/4) + 3(1/2) = -(2/4) + 3/2 = -(1/2) + 3/2 = 2/2 = 1\cr \end{array}
All of these check out, so our solution is correct.

 Math 15 (Spring 2005) web pages Created: 23 Mar 2005 Last modified: Jun 1, 2005 11:32:46 PM Comments to: dpvc@union.edu