Question

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

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

Answer 1

Answer: (A)

Let $\Delta=\begin{vmatrix}1 & a & a^{2}-b c \\ 1 & b & b^{2}-a c \\ 1 & c & c^{2}-a b\end{vmatrix}$
Applying $\left(R_{2} \rightarrow R_{2}-R_{1}, R_{3} \rightarrow R_{3}-R_{1}\right)$
$=\begin{vmatrix}1 & a & a^{2}-b c \\ 0 & b-a & (b-a)(a+b+c) \\ 0 & c-a & (c-a)(a+b+c)\end{vmatrix}$
$=(b-a)(c-a)\begin{vmatrix}1 & a & a^{2}-b c \\ 0 & 1 & a+b+c \\ 0 & 1 & a+b+c\end{vmatrix}=0(\because$ two rows are identical)