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 $\begin{pmatrix} 1 & 1 & 1 \\ a & b & c \\ ab & bc & ca \end{pmatrix}$.

Maybe it's unlikely that you would have learned of this method, but via Dodgson condensation, you can reduce this to a problem of computing the determinants of several $2\times2$ matrices, which you might find easier to compute strictly via row operations.
$$\begin{vmatrix}1&1&1\\a&b&c\\ab&bc&ca\end{vmatrix}=\begin{vmatrix}\dfrac{\begin{vmatrix}1&1\\a&b\end{vmatrix}}{b}&\dfrac{\begin{vmatrix}1&1\\b&c\end{vmatrix}}{b}\\[1ex]
\dfrac{\begin{vmatrix}a&b\\ab&bc\end{vmatrix}}{b}&\dfrac{\begin{vmatrix}b&c\\bc&ca\end{vmatrix}}{b}
\end{vmatrix}$$
The first determinant (in the $(1,1)$ position), for instance, would be
$$\begin{vmatrix}1&1\\a&b\end{vmatrix}=\begin{vmatrix}1&1\\0&b-a\end{vmatrix}=\begin{vmatrix}1&0\\0&b-a\end{vmatrix}=(b-a)\begin{vmatrix}1&0\\0&1\end{vmatrix}=b-a$$
