# Thread: [SOLVED] Is this system consistant?

1. ## [SOLVED] Is this system consistant?

$\displaystyle x_{1}-x_{2}+3x_{3}+2x_{4}=4$
$\displaystyle 2x_{1}-x_{2}+3x_{3}+x_{4}=6$
$\displaystyle 3x_{1}-2x_{2}+6x_{3}+3x_{4}=15$

When put into rref from i get
$\displaystyle \left(\begin{array}{cccc|c}1&0&0&-1&2\\0&1&-3&-3&-2\\0&0&0&0&5\\\hline\end{array}\right)$

Now this says 0 = 5, so presumably this system is inconsistant as 0 in my experiance doesn't equal 5.

Is there any way to find out if the system is inconsistant before i do 5 or more elementary row operations, which could be vital time wasted in an exam?

Thanks

2. Originally Posted by Rapid_W
$\displaystyle x_{1}-x_{2}+3x_{3}+2x_{4}=4$
$\displaystyle 2x_{1}-x_{2}+3x_{3}+x_{4}=6$
$\displaystyle 3x_{1}-2x_{2}+6x_{3}+3x_{4}=15$

When put into rref from i get
$\displaystyle \left(\begin{array}{cccc|c}1&0&0&-1&2\\0&1&-3&-3&-2\\0&0&0&0&5\\\hline\end{array}\right)$

Now this says 0 = 5, so presumably this system is inconsistant as 0 in my experiance doesn't equal 5.

Is there any way to find out if the system is inconsistant before i do 5 or more elementary row operations, which could be vital time wasted in an exam?

Thanks
Assuming your calculations are correct, your statement in red is correct. No further work is required.

3. Originally Posted by mr fantastic
Assuming your calculations are correct, your statement in red is correct. No further work is required.

I used maple and only elementary row operations, so it is correct.

But I was asking if it is possible to find out if the system is inconsistent before finding the rref form?

4. Originally Posted by Rapid_W
$\displaystyle x_{1}-x_{2}+3x_{3}+2x_{4}=4$
$\displaystyle 2x_{1}-x_{2}+3x_{3}+x_{4}=6$
$\displaystyle 3x_{1}-2x_{2}+6x_{3}+3x_{4}=15$

When put into rref from i get
$\displaystyle \left(\begin{array}{cccc|c}1&0&0&-1&2\\0&1&-3&-3&-2\\0&0&0&0&5\\\hline\end{array}\right)$

Now this says 0 = 5, so presumably this system is inconsistant as 0 in my experiance doesn't equal 5.

Is there any way to find out if the system is inconsistant before i do 5 or more elementary row operations, which could be vital time wasted in an exam?

Thanks
Why did you have to do 5 elementary operations? After the very first one, when you bring the first column (below the entry 1-1) to zero, the second and third rows are identical EXCEPT for the right most entry, which in the 2nd row is -2 and in the 3rd row is 3.
This already tells you the system is inconsistent since in the next step you will substract row 3 from row 2 and all the entries BUT the last one at the right in the 3rd row will vanish.

This says that 0*x1 + 0*x2 + 0*x3 + 0*x4 = 5, which of course is nonsense (unless you're working with a field of characteristic 5. If you don't understand this remark erase it right now from your brain: it isn't important for you right now)
Thus at most two steps, thought the result is visible already after the first one.

Tonio