Question

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

• A. $0$
• B. $a^3+b^3+c^3-3abc$
• C. $3abc$
• D. $(a+b+c{)}^3$

Answer 1

Answer: (A)

Let $\Delta =\left|\begin{array}{ccc}1& a& a^2-bc\\ 1& b& b^2-ac\\ 1& c& c^2-ab\end{array}\right|$
Applying $\left(R_2\to R_2-R_1,R_3\to R_3-R_1\right)$
$=\left|\begin{array}{ccc}1& a& a^2-bc\\ 0& b-a& (b-a)(a+b+c)\\ 0& c-a& (c-a)(a+b+c)\end{array}\right|$
$=(b-a)(c-a)\left|\begin{array}{ccc}1& a& a^2-bc\\ 0& 1& a+b+c\\ 0& 1& a+b+c\end{array}\right|=0(\because$ two rows are identical)