What is the simplest way to calculate this determinant using properties of determinants?

Question

A=$\begin{bmatrix} \frac ab & \frac bc & \frac ca\\a & b & c\\ab & bc & ca\end{bmatrix}$

Or maby there is no shorcut to calculate the det(A)?

Answer 1

Perform $C_3\to C_3-(c/b)C_2,C_2\to C_2-(b/a)C_1$, $$\sim\begin{vmatrix} \frac ab & \frac bc-1& \frac ca-1\\a &0 & 0\\ab & b(c-b) & c(a-c)\end{vmatrix}$$Now, factor out $a,(b-c),(c-a)$ from the first three columns respectively,$$\sim a(c-a)(b-c) \begin{vmatrix} \frac 1b & \frac 1c& \frac 1a\\1 &0 & 0\\b &-b& -c\end{vmatrix}$$and expand along the second row to get the answer $(a-b)(b-c)(c-a)$.