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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.


[HOME] Math 115 (Fall 2013) web pages
Created: 05 Sep 2013
Last modified: 05 Sep 2013 00:00:00
Comments to: dpvc@union.edu
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