JEE Math Practice Question

Let $\alpha, \beta$ and $\gamma$ be real numbers such that the system of linear equations \[\begin{gathered} x+2 y+3 z=\alpha \\ 4 x+5 y+6 z=\beta \\ 7 x+8 y+9 z=\gamma-1 \end{gathered}\] is consistent. Let $|M|$ represent the determinant of the matrix \[M=\left[\begin{array}{ccc} \alpha & 2 & \gamma \\ \beta & 1 & 0 \\ -1 & 0 & 1 \end{array}\right]\] Let $P$ be the plane containing all those $(\alpha, \beta, \gamma)$ for which the above system of linear equations is consistent, and $D$ be the square of the distance of the point $(0,1,0)$ from the plane $P$. What is the value of $|\mathrm{M}|$?