Question

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

(A) $a b c$

(B) $1 / a b c$

(C) $a b+b c+c a$

(D) 0

Answer 1

$\left|\begin{array}{lll}1 / a & a^2 & b c \\ 1 / b & b^2 & c a \\ 1 / c & c^2 & a b\end{array}\right|=\frac{1}{a b c}\left|\begin{array}{lll}1 & a^3 & a b c \\ 1 & b^3 & a b c \\ 1 & c^3 & a b c\end{array}\right|$

$=\frac{a b c}{a b c}\left|\begin{array}{lll}1 & a^3 & 1 \\1 & b^3 & 1 \\1 & c^3 & 1\end{array}\right|=0 \quad\left(\because \text { Columns } C_1 \text { and } C_2\right. \text { are same) }$

So, The correct option is (D).