I am a bit lost here, I created this overdetermined system from a traffic flow example and I am trying to figure out if it has a solution, and if so then if it's unique or not (which it won't be since its underdetermined). Here is my setup in matlab:
%A = x1 + 200 = 200 + x2 <=> x1 - x2 = 0
%B = x2 + x3 = x11 + 150 <=> x2 +x3 -x11 = 150
%C = 350+ x4 = 370+ x3 <=> -x3 + x4 = 20
%F = 410 + x12 = x4 + x5 <=> x4 +x5 -x12 = 410
%i = x5 + 330 = 510 +x6 <=> x5 -x6 = 180
%H = x6 + x7 = x9 + 210 <=> x6 + x7 -x9 = 210
%g = x8+300 = x7 + 380 <=> -x7 + x8 = 80
%d = 230+ x10 = x1+x8 <=> x1 +x8 -x10 = 230
%e = x9+x11 = x12+x10 <=> x9 - x10 +x11 -x12 = 0
%A) Below is the Matrix A and vector b for the underdetermined linear
%system above == not a unique solution to be found
A = [ 1 -1 0 0 0 0 0 0 0 0 0 0;
0 1 1 0 0 0 0 0 0 0 -1 0;
0 0 -1 1 0 0 0 0 0 0 0 0;
0 0 0 1 1 0 0 0 0 0 0 -1;
0 0 0 0 1 -1 0 0 0 0 0 0;
0 0 0 0 0 1 1 0 -1 0 0 0;
0 0 0 0 0 0 -1 1 0 0 0 0;
1 0 0 0 0 0 0 1 0 -1 0 0;
0 0 0 0 0 0 0 0 1 -1 1 -1];
b = [0; 150; 20; -410; 180; 210; 80; 230; 0];
The problem I am having understanding is when I use the reduced row echelon form of the augmented matrix I get:
AB = [A b]
rref(AB)
1 0 0 0 0 0 0 1 0 -1 0 0 0
0 1 0 0 0 0 0 1 0 -1 0 0 0
0 0 1 0 0 0 0 -1 0 1 -1 0 0
0 0 0 1 0 0 0 -1 0 1 -1 0 0
0 0 0 0 1 0 0 1 0 -1 1 -1 0
0 0 0 0 0 1 0 1 0 -1 1 -1 0
0 0 0 0 0 0 1 -1 0 0 0 0 0
0 0 0 0 0 0 0 0 1 -1 1 -1 0
0 0 0 0 0 0 0 0 0 0 0 0 1
From this, I read that the last line gives me x12 * 0 = 1, but that is impossible = so no solutions to the system.
But when I use linsolve(A,b) (A the coefficent matrix and b the output vector) I get a general solution to the system:
x = linsolve(A,b)
x =
213.3333
304.4444
-336.6667
-407.7778
88.8889
0
118.8889
107.7778
0
0
-91.1111
0
```