Find the determinant using colum or row operations I find problem in simplification. When I tried to simplify I ended up doing the regular process of finding the determinant value. The matrix is $\begin{pmatrix} 1 & 1 & 1 \\ a & b & c \\ ab & bc & ca \end{pmatrix}$.

$$\det\left(\begin{array}{ccc}1&1&1\\a&b&c\\ab&bc&ca\end {array}\right)=abc\det\left(\begin{array}{ccc}\frac{1}{a}&\frac{1}{b}&\frac{1}{c}\\1&1&1\\b&c&a\end {array}\right)=abc\det\left(\begin{array}{ccc}\frac{1}{a}-\frac{1}{b}&\frac{1}{b}&\frac{1}{c}-\frac{1}{b}\\0&1&0\\b-c&c&a-c\end {array}\right)=$$ 
$$abc\det\left(\begin{array}{ccc}\frac{1}{a}-\frac{1}{b}&\frac{1}{c}-\frac{1}{b}\\b-c&a-c\end {array}\right)=c(b-a)(a-c)-a(b-c)^{2}$$
