$\left|\begin{array}{ccc}\frac{1}{a}& a^2& bc\\ \frac{1}{b}& b^2& ca\\ \frac{1}{c}& c^3& ab\end{array}\right| =$

$$\begin{gathered} =\begin{array}{rcl} & & \left|\begin{array}{ccc}\frac{1}{\mathrm{a}}& {\mathrm{a}}^2& \mathrm{bc}\\ \frac{1}{\mathrm{b}}& {\mathrm{b}}^2& \mathrm{ca}\\ \frac{1}{\mathrm{c}}& {\mathrm{c}}^3& \mathrm{ab}\end{array}\right|\end{array}\begin{array}{rcl} & & \end{array} \\ =\left(\frac{1}{abc}\right)\begin{array}{rcl} & & \left|\begin{array}{ccc}\frac{1}{\mathrm{a}}\left(abc\right)& {\mathrm{a}}^2& \mathrm{bc}\\ \frac{1}{\mathrm{b}}\left(abc\right)& {\mathrm{b}}^2& \mathrm{ca}\\ \frac{1}{\mathrm{c}}\left(abc\right)& {\mathrm{c}}^3& \mathrm{ab}\end{array}\right|\end{array}\begin{array}{rcl} & & \end{array} \end{gathered}$$

$\begin{array}{rcl} & =& \left|\begin{array}{ccc}\mathrm{bc}& {\mathrm{a}}^2& \mathrm{bc}\\ \mathrm{ca}& {\mathrm{b}}^2& \mathrm{ca}\\ \mathrm{ab}& {\mathrm{c}}^2& \mathrm{ab}\end{array}\right|\end{array}$

$\begin{array}{rcl} & =& 0 \left(\therefore {\mathrm{c}}_1 \mathrm{and} {\mathrm{c}}_3 \mathrm{are} \mathrm{identical}\right)\end{array}$