Matrices And Determinants

Let \(\alpha\), \(\beta\) and \(\gamma\) be real numbers such that the system of linear equations

x + 2y + 3z = \(\alpha\)

4x + 5y + 6z = \(\beta\)

7x + 8y + 9z = \(\gamma\) \(-\) 1

is consistent. Let | M | represent the determinant of the matrix

\(M = \left[ {\matrix{ \alpha & 2 & \gamma \cr \beta & 1 & 0 \cr { - 1} & 0 & 1 \cr } } \right]\)

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.

The value of D is _________.