Let $A = [a_{ij}]$ be a $3\times 3$ matrix such that $\det(A) =-6$. If matrix $B$ is defined by $$B = \begin{bmatrix}3a_{33} & 3a_{32} & 3a_{31} \\ 2\left(a_{31}+a_{23}\right) & 2\left(a_{12}+a_{22}\right) & 2\left(a_{11}+a_{21}\right) \\ a_{23} & a_{22} & a_{21}\end{bmatrix}$$
evaluate $\det(B)$
$$\begin{align}\det(B) &= \left|\begin{array}{ccc}3a_{33} & 3a_{32} & 3a_{31} \\ 2\left(a_{13}+a_{23}\right) & 2\left(a_{12}+a_{22}\right) & 2\left(a_{11}+a_{21}\right) \\ a_{23} & a_{22} & a_{21}\end{array}\right|\\ &= (3)(2)\left|\begin{array}{ccc}a_{33} & a_{32} & a_{31} \\ \left(a_{13}+a_{23}\right) & \left(a_{12}+a_{22}\right) & \left(a_{11}+a_{21}\right) \\ a_{23} & a_{22} & a_{21}\end{array}\right| & \begin{pmatrix} R_1 \to \frac 13R_1 \\ R_2 \to \frac 12R_2\end{pmatrix} \\ &= 6\left|\begin{array}{ccc}a_{33} & a_{32} & a_{31} \\ a_{13} & a_{12} & a_{11} \\ a_{23} & a_{22} & a_{21}\end{array}\right| & \begin{pmatrix} R_2 \to R_2 - R_3\end{pmatrix} \\ &= -6\left|\begin{array}{ccc} a_{23} & a_{22} & a_{21} \\ a_{13} & a_{12} & a_{11} \\ a_{33} & a_{32} & a_{31}\end{array}\right| & \begin{pmatrix} R_1 \leftrightarrow R_3\end{pmatrix} \\ &= 6\left|\begin{array}{ccc} a_{13} & a_{12} & a_{11} \\ a_{23} & a_{22} & a_{21} \\ a_{33} & a_{32} & a_{31}\end{array}\right| & \begin{pmatrix} R_1 \leftrightarrow R_2\end{pmatrix} \\ &= -6\left|\begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right| & \begin{pmatrix} C_1 \leftrightarrow C_3\end{pmatrix} \\ &= -6\det(A) \\ &= 36\end{align}$$