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. (b - c)
• C. abc
• D. Both b and c

Answer 1

The correct option is B (b - c)

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

$R_2\to (R_2-R_1);R_3\to (R_3-R_1)$

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

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

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

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