Question

Evaluate the following :

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

Answer 1

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

Applying $R_2\to R_2-R_1,R_3\to R_3-R_1,$ we get

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

$\Rightarrow \Delta =(a-b)(a-c)\left|\begin{array}{ccc}1& a& bc\\ 0& -1& c\\ 0& -1& b\end{array}\right|$ [Taking (a-b) and (a-c)common from $R_1$ and $R_3$ respectively.]

$\Rightarrow \Delta =(a-b)(a-c)(-b+c)$ [Expanding along first column]

$\Rightarrow \Delta =(a-b)(b-c)(c-a)$