Math 115 (Notes)

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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.

 Math 115 (Fall 2011) web pages Created: 02 Sep 2011 Last modified: Sep 3, 2011 9:32:49 AM Comments to: dpvc@union.edu