Question

Prove that $\left|\begin{array}{ccc}1& a& bc\\ 1& b& ca\\ 1& c& ab\end{array}\right|=(a-b)(b-c)(c-a)$.

Answer 1

Now,

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

[$R_2^{\prime }=R_2-R_1$ and $R_3^{\prime }=R_3-R_1$]

$=\left|\begin{array}{ccc}1& a& bc\\ 0& b-a& c(a-b)\\ 0& c-a& b(a-c)\end{array}\right|$

$=(b-a)(c-a)\left|\begin{array}{ccc}1& a& bc\\ 0& 1& -c\\ 0& 1& -b\end{array}\right|$

$=(b-a)(c-a)(c-b)$

$=(a-b)(b-c)(c-a)$.