One solution ofanyhomogeneous system is the "0" solution: all unknowns equal to 0. If there exist other solutions, then the determinant of the coefficient matrix is not 0 (equivalently, the coefficient matrix is not invertible). That means that, if we were to row reduce the coefficient matrix, we would have at least one row of all "0"s. Certainly, it would be possible to have a "right side" such that the last column of that row, after row reducing the augmented matrix, would NOT be 0.