Consider the following system of linear equations: \begin{eqnarray} { {x_1 + 2x_2 - x_3 = 1 }\\ {x_2 + \alpha x_3 = 0} \\ {2x_1 + 2x_2 + x_3 = \beta}} \end{eqnarray}
I have the row reduced matrix below but I end up with a pivot position in every row so I'm not sure how to fine values of $\alpha$ and $\beta$ that give infinite solutions.
\begin{bmatrix} 1 & 0 & -1-2 \alpha & 1 \\ 0 & 1 & \alpha & 0 \\ 0 & 0 & 1 & \frac{\beta - 1}{2} \end{bmatrix}
Please let me know what I'm doing/I've done wrong.