Question

Prove that

$\left|\begin{array}{ccc}1& ab& \frac{1}{a}+\frac{1}{b}\\ 1& bc& \frac{1}{b}+\frac{1}{c}\\ 1& ca& \frac{1}{c}+\frac{1}{a}\end{array}\right|$=0

Answer 1

$\frac{1}{a}+\frac{1}{b}=\frac{a+b}{ab}=\frac{c(a+b)}{abc}$

$\therefore △=\frac{1}{abc}$

$\left|\begin{array}{ccc}1& ab& bc+ca\\ 1& bc& ca+ab\\ 1& ca& ab+bc\end{array}\right|$ =0

by $C_3+C_2$ and taking $\sum \left(ab\right)$ out thus making $C_1$ and $C_3$ identical.