I try to find the solutions for the following expression with the Gauß Formula:
$$1x_1-1x_2+3x_3=0$$
$$2x_1 + 3x_2 - 1x_3 = 0$$
$$3x_1+7x_2-5x_3 =0$$
So i started:
$ \begin{pmatrix} 1 & -1 & 3 \\ 2 & 3 & -1 \\ 3 & 7 & -5 \\ \end{pmatrix} $ Next i performed: $I -2* II $ = $ \begin{pmatrix} 1 & -1 & 3 \\ 0 & 5 & -7 \\ 0 & 10 & -14 \\ \end{pmatrix} $
Now my problem started, when i count $-0.5*III + II$ the whole last row gets $0$
$$\begin{pmatrix} 1 & -1 & 3 \\ 0 & 5 & -7 \\ 0 & 0 & 0 \\ \end{pmatrix}$$
Did i something wrong? How should i continue? Thanks
If the whole row hadn't become 0, then the only solution would be the trivia solution $x_1 = x_2 = x_3 = 0$.
What you've done is correct. Now you can solve this by back-substitution. The system now is: $x_1 - x_2 + 3x_3 = 0\\ 5x_2 - 7x_3 = 0$
You can find expressions for $x_1$ and $x_2$ in terms of $x_3$, or assume a particular value for $x_3$ and solve for $x_1$ and $x_2$.