Question

The value of $\left|\begin{array}{ccc}1& 1& 1\\ a& b& c\\ a^2& b^2& c^2\end{array}\right|$ is

• A. $(a+b)(b+c)(c+a)$
• B. $(a-b)(b-c)(c-a)$
• C. $a^3+b^3+c^{3}3abc$
• D. $\left(ab\right)\left(bc\right)\left(ac\right)$

Answer 1

Answer: (B)

$(a-b)(b-c)(c-a)$

Let $\Delta =\left|\begin{array}{ccc}1& 1& 1\\ a& b& c\\ a^2& b^2& c^2\end{array}\right|$

Applying $C_2\to C_2-C_1;C_3\to C_3-C_1$

$\Delta =\left|\begin{array}{ccc}1& 0& 0\\ a& b-a& c-a\\ a^2& b^2-a^2& c^2-a^2\end{array}\right|$

$=(b-a)(c^2-a^2)-(c-a)(b^2-a^2)=(b-a)(c-a)(c+a)-(b-a)(b+a)(c-a)=(b-a)(c-a)(c-b)$