Determinant of Matrix M for Consistent System of Linear Equations

To determine the value of $|M|$, let us first find the condition for the given system of linear equations to be consistent.

The system of linear equations is given by: $x + 2y + 3z = \alpha \quad \text{--- (1)}$ $4x + 5y + 6z = \beta \quad \text{--- (2)}$ $7x + 8y + 9z = \gamma - 1 \quad \text{--- (3)}$

Observe the linear combination of the left-hand sides of equations $(1)$ and $(3)$: $(x + 2y + 3z) + (7x + 8y + 9z) = 8x + 10y + 12z = 2(4x + 5y + 6z)$

For the system of linear equations to be consistent, the constant terms must satisfy the exact same linear relationship: $\alpha + (\gamma - 1) = 2\beta$ $\alpha - 2\beta + \gamma - 1 = 0$ $\alpha - 2\beta + \gamma = 1 \quad \text{--- (4)}$

Now, we calculate the determinant of matrix $M$:

$$M = \begin{bmatrix} \alpha & 2 & \gamma \\ \beta & 1 & 0 \\ -1 & 0 & 1 \end{bmatrix}$$

Expanding the determinant along the third row: $|M| = -1 \cdot (2 \cdot 0 - 1 \cdot \gamma) - 0 + 1 \cdot (\alpha \cdot 1 - 2 \cdot \beta)$ $|M| = \gamma + \alpha - 2\beta$ $|M| = \alpha - 2\beta + \gamma$

Substituting the consistency condition from equation $(4)$: $|M| = 1$