Question

If $\Delta =\left|\begin{array}{ccc}1& a& bc\\ 1& b& ca\\ 1& c& ab\end{array}\right|$, then which of the following is a factor of $\Delta$

• A. $a+b$
• B. a - b
• C. $abc$
• D. a+b+c

Answer 1

The correct option is B a - b

$\Delta =\left|\begin{array}{ccc}1& a& bc\\ 1& b& ca\\ 1& c& ab\end{array}\right|$
$R_1\to R_1-R_2$
$\Delta =\left|\begin{array}{ccc}0& a-b& bc-ca\\ 1& b& ca\\ 1& c& ab\end{array}\right|$
Take out common (a-b) from $R_1$
$\Delta =(a-b)\left|\begin{array}{ccc}0& 1& -c\\ 1& b& ca\\ 1& c& ab\end{array}\right|$
$\therefore$ (a-b) is a factor of $\Delta$.