Question #40856

Let $A, B, C, D$ be four points in $\mathbb{R}^3$ such that the volume of the tetrahedron $ABCD$ is $7$. Let $M$ be the $4 \times 4$ matrix defined by $M = \begin{pmatrix} \vec{a} & \vec{b} & \vec{c} & \vec{d} \\ 1 & 1 & 1 & 1 \end{pmatrix}$ where $\vec{a}, \vec{b}, \vec{c}, \vec{d}$ are the position vectors of $A, B, C, D$ respectively, and $\det(M) > 0$. Let $\vec{g}$ be the position vector of the centroid of the tetrahedron $ABCD$. Let $N$ be a $4 \times 4$ matrix whose columns are $C_1, C_2, C_3, C_4$, where $C_1 = \begin{pmatrix} 3\vec{a} - \vec{g} \\ 2 \end{pmatrix}, \quad C_2 = \begin{pmatrix} 3\vec{b} - \vec{g} \\ 2 \end{pmatrix}, \quad C_3 = \begin{pmatrix} 3\vec{c} - \vec{g} \\ 2 \end{pmatrix}, \quad C_4 = \begin{pmatrix} 3\vec{d} - \vec{g} \\ 2 \end{pmatrix}$ If $P = \frac{1}{3} N$, what is the value of $\det(P)$?