Solution
= Solution
Let $D_\beta=\operatorname{diag}(\beta_1,\ldots,\beta_n)$ and $D_\gamma=\operatorname{diag}(\gamma_1,\ldots,\gamma_n)$. The specified entries mean $C=D_\beta A D_\gamma$. Multiplication of <determinants> therefore gives
$$
\boxed{|C|=\left(\prod_{i=1}^n\beta_i\right)\left(\prod_{j=1}^n\gamma_j\right)|A|}.
$$
This is also the direct row- and column-scaling rule, and it includes zero scaling factors.