Question #40897

Let $N$ be a $3 \times 3$ real matrix such that $N \neq O$ and $N^2 = O$. Let $P$ be the matrix obtained by swapping the $2^{\text{nd}}$ and $3^{\text{rd}}$ rows of the identity matrix $I_3$, and $Q$ be the matrix obtained by swapping the $1^{\text{st}}$ and $2^{\text{nd}}$ columns of $I_3$. A tetrahedron $T$ has vertices at the origin $O(0,0,0)$ and the points with position vectors given by the columns of the matrix $M = P(I - N)^{-1}Q$. Let $D$ be the scalar triple product of the column vectors of $M$ taken in order from left to right, and $G(x, y, z)$ be the centroid of $T$. If $N \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 4 \\ -2 \\ 6 \end{bmatrix}$, find the value of $D + 4(x + y + z)$.