Question

Using properties of determinant solve :-
$\left|\begin{array}{ccc}1& a& a^2-bc\\ 1& b& b^2-ac\\ 1& c& c^2-ab\end{array}\right|=$

Answer 1

$\left|\begin{array}{ccc}1& a& a^2-bc\\ 1& b& b^2-ac\\ 1& c& c^2-ab\end{array}\right|$

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

$\left|\begin{array}{ccc}0& a-b& (a^2-b^2)+c(a-b)\\ 0& b-c& (b^2-c^2)+a(b-c)\\ 1& c& c^2-ab\end{array}\right|$

Expanding the determinant

$1\left[\right(a-b\left)\right[(b-c)(a+b+c)]-(b-c\left)\right[(a-b)(a+b+c)\left]\right]$

$\Rightarrow 0$.