For which vectors b1 b2 b3 do the following systems Ax b ha

For which vectors (b_1, b_2, b_3) do the following systems Ax = b have a solution?

Solution

Matlab code:

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syms x1 x2 x3 b1 b2 b3

A1 = [1 1 1; 0 1 1; 0 0 1];
A2 = [1 1 1; 0 1 1; 0 0 0];
A3 = [1 1 1; 0 0 1; 0 0 1];

B1 = inv(A1);
B2 = inv(A2);
B3 = inv(A3);

S1 = solve(A1*[x1;x2;x3]==[b1;b2;b3],x1,x2,x3); % this system has solution.
S2 = solve(A2*[x1;x2;x3]==[b1;b2;b3],x1,x2,x3);
S3 = solve(A3*[x1;x2;x3]==[b1;b2;b3],x1,x2,x3);

% note that the solutions S2 and S3 are empty because the system has no
% solution.

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Handwritting solution: (I could not upload the images to this plattform, sorry)

https://drive.google.com/file/d/0B5Kioj2EbCtBNXA5SENqbFdJNUk/view

https://drive.google.com/open?id=0B5Kioj2EbCtBbVZEZDRoRWFxdEU

Federico.