Homework Help: Gaussian Elim Problem

1. Jun 27, 2009

1. The problem statement, all variables and given/known data
Examine the solutions of

$$\left[\begin{array}{c}x_1-x_2+x_3=c\\2x_1-3x_2+4x_3=0\\3x_1-4x_2+5x_3=1\end{array}\right]$$

$$\text{when }c=1\text{ and }c\ne1$$

3. The attempt at a solution

$$\left[\begin{array}{cccc}1& -1& 1& c\\2& -3& 4& 0\\3& -4& 5& 1\end{array}\right]$$

$$\left[\begin{array}{cccc}1& -1& 1& c\\0& -1& 2& -2c\\0& -1& 2& -2c\end{array}\right]$$

$$\left[\begin{array}{cccc}1& -1& 1& c\\0& 1& -2& 2c\\0& 0& 0& 0\end{array}\right]$$

What is the next step here?

2. Jun 27, 2009

Pengwuino

What this step is telling you is that x3 is arbitrary. Continue elimination if you want, but you can construct your solution from this. For example, the 2nd row shows that x2 = 2c +2x3.

3. Jun 27, 2009

snipez90

The entry at the far bottom right of the second matrix should be 1 - 3c.

4. Jun 27, 2009

Pengwuino

oops yah I didn't notice that. it is 1-3c which changes the situation...

5. Jun 27, 2009

Doh!!!!!

edit: Hey!! PF changed my DOH from all Caps!

6. Jun 27, 2009

$$\left[\begin{array}{cccc}1& -1& 1& c\\0& -1& 2& -2c\\0& -1& 2& 1-2c\end{array}\right]$$

$$\left[\begin{array}{cccc}1& -1& 1& c\\0& 1& -2& 2c\\0& 0& 0& 1\end{array}\right]$$

Now.... have I made another silly error?

Is the elimination finished?

7. Jun 27, 2009

Pengwuino

No, you made another mistake. You're subtracting 3 times the first row so you should have 1-3c as the bottom right coefficient, not 1-2c.

8. Jun 27, 2009

$$\left[\begin{array}{cccc}1& -1& 1& c\\0& 1& -2& 2c\\0& 0& 0& (1-c)\end{array}\right]$$
c$\ne$1-->inconsistent eqs