The matrix AAA is 6×46 \times 46×4 with rank 3. For which values of b\mathbf{b}b is the system Ax=bA\mathbf{x} = \mathbf{b}Ax=b consistent?
For all b\mathbf{b}b, since there are more equations than unknowns
Only when b\mathbf{b}b lies in the column space of AAA, which is 3-dimensional
Only when b\mathbf{b}b is orthogonal to the null space of AAA
Only when rank([A∣b])=4([A|\mathbf{b}]) = 4([A∣b])=4