[SOLVED] Gauss-Jordan method

**wilsoac** · February 18th 2009, 03:18 PM

Can someone please explain to me the Guass-Jordan method to solve a system of equations?

My homework question is:

Use the Gauss-Jordan method to solve the system of equations.
4x-6y=54
20x-30y=270

I am sure that if someone would simplify the method for me (I guess explain it to me a little more clearly) I would be able to figure it out. I don't want the answer, I would like to figure it out on my own, but just need someone to lead me in the right direction.

**javax** · February 18th 2009, 04:03 PM

Hello.
Using the Gauss-Jordan method you'll have to transform the augmented matrix using row elimination to transform the matrix into this form.
$ \begin{pmatrix} a_{11} & 0 & \dots & 0 & \vdots & a_{1, n+1} \\ 0 & a_{22} & \ddots & \vdots & \vdots & a_{2, n+1}\\ \vdots & \ddots & \ddots & 0 & \vdots & \vdots \\ 0 & \dots & 0 & a_{nn} & \vdots & a_{n, n+1} \end{pmatrix}$

and the solutions will be
$x_i = \frac{a_{i, n+1}}{a_{ii}}$

here's one example

$2x_1-x_2=4$
$x_1-x_2=1$

the augmented matrix is

$\begin{pmatrix} 2 & -1 & \vdots & 4 \\ 1 & -1 & \vdots & 1 \end{pmatrix}$

after performing these transformations:Row2-(1/2)*Row1 then after this, Row1-2*Row2
the augmented matrix will be
$ \begin{pmatrix} 2 & 0 & \vdots & 6 \\ 0 & -\frac{1}{2} & \vdots & -1 \end{pmatrix}$

the solutions will be: $x_1=\frac{6}{2} = 3, x_2=\frac{-1}{\frac{-1}{2}}=2$

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