Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 2 / 17 / b / iii / Solution

Let $D_{\beta }=\mathrm{diag}({\beta }_1,\dots ,{\beta }_n)$ and $D_{\gamma }=\mathrm{diag}({\gamma }_1,\dots ,{\gamma }_n)$. The specified entries mean $C=D_{\beta }A{D}_{\gamma }$. Multiplication of determinants therefore givesThis is also the direct row- and column-scaling rule, and it includes zero scaling factors.