Dear ladies and gentlemen,
over time I noticed I (and other) again and again have problems solving "systems of linear equations". It seems depending of the steps one chooses, we get different results!! How can that be? Should one not always get the same results no matter which path he goes down? What am I missing? Are there maybe rules I don't know and use even though I shouldn't?
I want to give you an example. The set of equations is from a state-price security calculation (see "State Preference Approach") which we shall solve using the Gaussian elimination:
I: 2P(1) + 2P(2) + 2P(3) =1,6
II: 3P(1) + 0P(2) + 1P(3) =1,0
III: 0P(1) + 2P(2) + 1P(3) =0,8
The solution shall be: P(1) = 0,2 P(2)=0,2 and P(3)=0,4
Not only got I different numbers on the first try that totally went in "into space", I want to write down for you my second approach to check for mistakes:
II-III = IV = 3P(1) - 2P(2) = 0,2 --> 2P(2) = 3P(1) - 0,2 (this far I'm with the solution) then I simply plug in the result in the following lines:
in III: 4P(1) = 1 --> P(1) = 0,25
in II: 0,75 + P(3) = 1,0 --> P(3) = 0,25
in I: 0,5 + 2P(2) + 0,5 = 1,6 --> P(2) = 0,3
But these results seem to differ. So it seems I clearly miss some important rule!
Must I not "plug in" results into other rows, as the "Gaussian" system seems to avoid? But how can there be a difference / can I be forced to avoid "plugging in" results?
This should normally be allowed, shouldn't it? Can you help me find my blind spot?
Thanks for your help in advance.