Question

Without expanding, show that the value of the following determinant is zero:

\[\begin{vmatrix}1/a & a^2 & bc \\ 1/b & b^2 & ac \\ 1/c & c^2 & ab\end{vmatrix}\]

Answer 1

\[ ∆ = \begin{vmatrix}\frac{1}{a} & a^2 & bc \\ \frac{1}{b} & b^2 & ac \\ \frac{1}{c} & c^2 & ab\end{vmatrix}\]
\[ = \begin{vmatrix}1 & a^3 & abc \\ 1 & b^3 & abc \\ 1 & c^3 & abc\end{vmatrix} \left[ \text{ Applying } R_1 \to a R_1 , R_2 \to b R_2 and R_3 \to c R_3 \right]\]
\[ = abc\begin{vmatrix}1 & a^3 & 1 \\ 1 & b^3 & 1 \\ 1 & c^3 & 1\end{vmatrix}\]
\[ \Rightarrow ∆ = 0\]