Question

Find a 3  3-matrix A such that there exists
b1 =vector(0) and
b2 =vector(0), such that
vector(A)x = b1 has innitely many solutions, but vector(A)x = b2 has no solutions.

Answer 1

$$A = \left( \begin{array}{ccc} 1 & 2 & 3 \\ 3 & 1 & 2 \\ 4 & 3 & 5 \end{array} \right), b_0 = \left( \begin{array}{c} 1 \\ 1 \\ 2 \end{array} \right), b_1 = \left( \begin{array}{c} 1 \\ 1 \\ 1 \end{array} \right)$$ $$Ax = b_0$$ $$Ax = b_1$$

First system has infinitely many solutions, second has no solutions. (rank(A|b_0)=2=rank(A), and rank(A|b_1)=3 not equal to rank(A)).